Class 12 Physics Chapter 4: Orthographic Projection of Points and Lines NCERT Solutions
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Complete Chapter Solutions
A point \(D\) is 25 mm from H.P. and 30 mm from V.P. Draw its projections considering it in first and third quadrants.
Concept used. The front view records distance from H.P.; the top view records distance from V.P. In the first quadrant, front view is above \(XY\) and top view is below \(XY\). In the third quadrant, front view is below \(XY\) and top view is above \(XY\).
- Draw one horizontal reference line \(XY\).
- At a convenient point, draw a thin projector perpendicular to \(XY\).
- For the first quadrant, mark \(d'\) 25 mm above \(XY\) and \(d\) 30 mm below \(XY\) on the same projector.
- For the third quadrant, draw another projector and mark \(d'\) 25 mm below \(XY\) and \(d\) 30 mm above \(XY\).
- Label the two sets clearly as I quadrant and III quadrant.
Drawing check
The two views for each case must lie on the same projector.Construction route. Use the sign rule: height above H.P. controls the front view and distance in front of or behind V.P. controls the top view.
- Draw one horizontal reference line \(XY\).
- At a convenient point, draw a thin projector perpendicular to \(XY\).
- For the first quadrant, mark \(d'\) 25 mm above \(XY\) and \(d\) 30 mm below \(XY\) on the same projector.
- For the third quadrant, draw another projector and mark \(d'\) 25 mm below \(XY\) and \(d\) 30 mm above \(XY\).
- Label the two sets clearly as I quadrant and III quadrant.
A point \(P\) is 15 mm above H.P. and 20 mm in front of V.P. Another point \(Q\) is 25 mm behind V.P. and 40 mm below H.P. Draw the projections.
Concept used. A point above H.P. gives front view above \(XY\). A point below H.P. gives front view below \(XY\). A point in front of V.P. gives top view below \(XY\), while a point behind V.P. gives top view above \(XY\).
- Draw \(XY\) and two projectors.
- For \(P\), mark \(p'\) 15 mm above \(XY\) and \(p\) 20 mm below \(XY\).
- For \(Q\), mark \(q'\) 40 mm below \(XY\) and \(q\) 25 mm above \(XY\).
- Keep each front view and top view on its own projector.
- Darken only the final point marks and labels.
Projection tip
\(P\) has views on opposite sides of \(XY\) in the first-quadrant arrangement; \(Q\) has the third-quadrant arrangement.Construction route. Classify each point first. Once the quadrant is known, the plotting becomes direct.
- Draw \(XY\) and two projectors.
- For \(P\), mark \(p'\) 15 mm above \(XY\) and \(p\) 20 mm below \(XY\).
- For \(Q\), mark \(q'\) 40 mm below \(XY\) and \(q\) 25 mm above \(XY\).
- Keep each front view and top view on its own projector.
- Darken only the final point marks and labels.
Draw the projections of points \(B\), \(D\), \(E\) and \(F\) on the same \(XY\): \(B\) is 20 mm above H.P. and 25 mm in front of V.P.; \(D\) is 25 mm below H.P. and 15 mm behind V.P.; \(E\) is 15 mm above H.P. and 10 mm behind V.P.; \(F\) is 20 mm below H.P. and 25 mm in front of V.P.
Concept used. Each point is located by two signed distances: height from H.P. for the front view and distance from V.P. for the top view.
- Draw one common \(XY\) line.
- Draw four separate thin projectors for \(B\), \(D\), \(E\) and \(F\).
- Mark \(b'\) 20 mm above \(XY\) and \(b\) 25 mm below \(XY\).
- Mark \(d'\) 25 mm below \(XY\) and \(d\) 15 mm above \(XY\).
- Mark \(e'\) 15 mm above \(XY\) and \(e\) 10 mm above \(XY\).
- Mark \(f'\) 20 mm below \(XY\) and \(f\) 25 mm below \(XY\).
- Label every front view with a dash and every top view without a dash.
Drawing check
Second and fourth quadrant points have both views on the same side of \(XY\).Construction route. Make a small quadrant table before plotting. It prevents view placement errors.
- Draw one common \(XY\) line.
- Draw four separate thin projectors for \(B\), \(D\), \(E\) and \(F\).
- Mark \(b'\) 20 mm above \(XY\) and \(b\) 25 mm below \(XY\).
- Mark \(d'\) 25 mm below \(XY\) and \(d\) 15 mm above \(XY\).
- Mark \(e'\) 15 mm above \(XY\) and \(e\) 10 mm above \(XY\).
Line \(CD\) is in V.P., parallel to H.P. and 40 mm long. End \(C\) is 30 mm above H.P. Draw its projections.
Concept used. A line in V.P. and parallel to H.P. shows true length in front view. Its top view lies on \(XY\) because every point is on V.P.
- Draw \(XY\).
- Draw \(c'd'\) 40 mm long, parallel to \(XY\), and 30 mm above it.
- Drop projectors from \(c'\) and \(d'\) to \(XY\).
- Mark \(c\) and \(d\) on \(XY\) at the projector feet.
- Join \(c\) to \(d\) on \(XY\).
Projection tip
The front view is true length and the top view coincides with \(XY\).Construction route. A V.P. line has zero distance from V.P.; that is why the top view sits on \(XY\).
- Draw \(XY\).
- Draw \(c'd'\) 40 mm long, parallel to \(XY\), and 30 mm above it.
- Drop projectors from \(c'\) and \(d'\) to \(XY\).
- Mark \(c\) and \(d\) on \(XY\) at the projector feet.
- Join \(c\) to \(d\) on \(XY\).
Line \(EF\) is parallel to V.P., 25 mm in front of V.P., and lies in H.P. Its length is 40 mm. Draw its projections.
Concept used. A line in H.P. has front view on \(XY\). If it is parallel to V.P., its top view shows true length.
- Draw \(XY\).
- Draw \(ef\) 40 mm long, parallel to \(XY\), and 25 mm below it.
- Project upward from \(e\) and \(f\) to \(XY\).
- Mark \(e'\) and \(f'\) on \(XY\).
- Join \(e'f'\) on the reference line.
Drawing check
The top view is true length and the front view lies on \(XY\).Construction route. For a line in H.P., all points have zero height, so the front view is on \(XY\).
- Draw \(XY\).
- Draw \(ef\) 40 mm long, parallel to \(XY\), and 25 mm below it.
- Project upward from \(e\) and \(f\) to \(XY\).
- Mark \(e'\) and \(f'\) on \(XY\).
- Join \(e'f'\) on the reference line.
Line \(GH\) is in both H.P. and V.P. and is 40 mm long. Draw its projections.
Concept used. A line common to H.P. and V.P. lies on their intersection, the reference line \(XY\).
- Draw \(XY\).
- Mark a 40 mm segment on \(XY\).
- Label the same segment as \(g'h'\) for front view.
- Also label it as \(gh\) for top view.
- Keep the drawing neat because the two views overlap exactly.
Projection tip
Both views are on \(XY\) and have the same length.Construction route. This is the simplest case because the line has zero height and zero distance from V.P.
- Draw \(XY\).
- Mark a 40 mm segment on \(XY\).
- Label the same segment as \(g'h'\) for front view.
- Also label it as \(gh\) for top view.
- Keep the drawing neat because the two views overlap exactly.
Line \(JK\) is perpendicular to H.P. and 20 mm in front of V.P. The nearest point from H.P. is \(J\), which is 15 mm above H.P. Draw projections for a 40 mm line.
Concept used. A line perpendicular to H.P. appears as a point in top view and shows true length in front view.
- Draw \(XY\) and one projector.
- Mark \(j'\) 15 mm above \(XY\).
- From \(j'\), draw \(j'k'\) vertical and 40 mm long upward.
- On the same projector, mark the top view point \(j,k\) 20 mm below \(XY\).
- Label the coincident top view as \(j(k)\).
Drawing check
The top view is a point because the line is seen end-on from above.Construction route. True length appears in the view parallel to the line; here that is the front view.
- Draw \(XY\) and one projector.
- Mark \(j'\) 15 mm above \(XY\).
- From \(j'\), draw \(j'k'\) vertical and 40 mm long upward.
- On the same projector, mark the top view point \(j,k\) 20 mm below \(XY\).
- Label the coincident top view as \(j(k)\).
Line \(UV\) is perpendicular to V.P. The farthest end \(V\) is 65 mm in front of V.P. and 20 mm above H.P. Draw projections for a 40 mm line.
Concept used. A line perpendicular to V.P. appears as a point in front view and shows true length in top view.
- Draw \(XY\) and one projector.
- Mark the coincident front view \(u'v'\) 20 mm above \(XY\).
- Mark \(v\) 65 mm below \(XY\).
- Since the line is 40 mm long and perpendicular to V.P., mark \(u\) 25 mm below \(XY\) on the same projector.
- Join \(u\) to \(v\) as the top view.
Projection tip
\(uv = 40\) mm and \(u'v'\) is a point.Construction route. Distance from V.P. is measured in top view; use it to place the nearer and farther ends.
- Draw \(XY\) and one projector.
- Mark the coincident front view \(u'v'\) 20 mm above \(XY\).
- Mark \(v\) 65 mm below \(XY\).
- Since the line is 40 mm long and perpendicular to V.P., mark \(u\) 25 mm below \(XY\) on the same projector.
- Join \(u\) to \(v\) as the top view.
Line \(AB\) is parallel to both H.P. and V.P., 25 mm behind V.P. and 30 mm below H.P. Its length is 40 mm. Draw projections.
Concept used. A line parallel to both planes shows true length in both views. Behind V.P. puts top view above \(XY\), and below H.P. puts front view below \(XY\).
- Draw \(XY\).
- Draw \(a'b'\) 40 mm long and parallel to \(XY\), 30 mm below it.
- Draw projectors from \(a'\) and \(b'\).
- On those projectors, mark \(a\) and \(b\) 25 mm above \(XY\).
- Join \(ab\) parallel to \(XY\).
Drawing check
Both views remain true length and parallel to \(XY\).Construction route. Parallel to both planes means no foreshortening in either view.
- Draw \(XY\).
- Draw \(a'b'\) 40 mm long and parallel to \(XY\), 30 mm below it.
- Draw projectors from \(a'\) and \(b'\).
- On those projectors, mark \(a\) and \(b\) 25 mm above \(XY\).
- Join \(ab\) parallel to \(XY\).
Line \(LM\) is 30 mm behind V.P. and perpendicular to H.P. The nearest point from H.P. is \(L\), 10 mm above H.P. Length is 40 mm. Draw projections.
Concept used. A vertical line perpendicular to H.P. has a point top view. Behind V.P. places that top view above \(XY\).
- Draw a projector.
- Mark \(l'\) 10 mm above \(XY\).
- Draw \(l'm'\) 40 mm upward as the front view.
- Mark coincident top view \(l(m)\) 30 mm above \(XY\).
- Label clearly to show the top view is one point.
Projection tip
The top view must be above \(XY\) because the line is behind V.P.Construction route. For a vertical line, only the height changes; the distance from V.P. stays constant.
- Draw a projector.
- Mark \(l'\) 10 mm above \(XY\).
- Draw \(l'm'\) 40 mm upward as the front view.
- Mark coincident top view \(l(m)\) 30 mm above \(XY\).
- Label clearly to show the top view is one point.
Line \(NP\) is 30 mm below H.P. and perpendicular to V.P. The nearest point from V.P. is \(P\), 10 mm in front of V.P. Length is 40 mm. Draw projections.
Concept used. A line perpendicular to V.P. has a point front view. Since it is below H.P., that front view is below \(XY\).
- Draw one projector.
- Mark \(p'\) and \(n'\) coincident 30 mm below \(XY\).
- Mark \(p\) 10 mm below \(XY\).
- From \(p\), mark \(n\) 40 mm farther from \(XY\), so \(n\) is 50 mm below \(XY\).
- Join \(pn\) as the top view.
Drawing check
The top view carries the true length.Construction route. Nearest to V.P. means the smaller distance from \(XY\) in top view.
- Draw one projector.
- Mark \(p'\) and \(n'\) coincident 30 mm below \(XY\).
- Mark \(p\) 10 mm below \(XY\).
- From \(p\), mark \(n\) 40 mm farther from \(XY\), so \(n\) is 50 mm below \(XY\).
- Join \(pn\) as the top view.
Line \(QR\) is 10 mm below H.P. and perpendicular to V.P. The farthest point from V.P. is \(Q\), 65 mm behind V.P. Length is 40 mm. Draw projections.
Concept used. Behind V.P. places the top view above \(XY\). A line perpendicular to V.P. is true length in top view and a point in front view.
- Draw one projector.
- Mark \(q'\) and \(r'\) coincident 10 mm below \(XY\).
- Mark \(q\) 65 mm above \(XY\).
- Since \(QR = 40\) mm and \(Q\) is farthest behind V.P., mark \(r\) at 25 mm above \(XY\).
- Join \(qr\).
Projection tip
The top view length \(qr\) must be 40 mm.Construction route. When both ends are behind V.P., both top-view points stay above \(XY\).
- Draw one projector.
- Mark \(q'\) and \(r'\) coincident 10 mm below \(XY\).
- Mark \(q\) 65 mm above \(XY\).
- Since \(QR = 40\) mm and \(Q\) is farthest behind V.P., mark \(r\) at 25 mm above \(XY\).
- Join \(qr\).
Line \(ST\) is perpendicular to H.P. and behind V.P. The nearest point from H.P. is \(S\), which is 20 mm from V.P. and 15 mm below H.P. Length is 40 mm. Draw projections.
Concept used. A line perpendicular to H.P. shows true length in front view and appears as a point in top view.
- Draw one projector.
- Mark \(s'\) 15 mm below \(XY\).
- Draw \(s't'\) 40 mm vertically from \(s'\).
- Because the line is behind V.P., mark \(s(t)\) 20 mm above \(XY\).
- Label the coincident top view carefully.
Drawing check
The top view is above \(XY\) and remains a single point.Construction route. Do not move the top view while the front view extends. Only height changes.
- Draw one projector.
- Mark \(s'\) 15 mm below \(XY\).
- Draw \(s't'\) 40 mm vertically from \(s'\).
- Because the line is behind V.P., mark \(s(t)\) 20 mm above \(XY\).
- Label the coincident top view carefully.
A thin pentagonal plate of 35 mm sides is inclined at \(30^\circ\) to H.P. and perpendicular to V.P. One edge is perpendicular to V.P., 20 mm above H.P., and its nearer end is 30 mm in front of V.P. Draw the projections.
Concept used. A plane figure shows true shape only in the view parallel to its surface. If the plane is inclined to one reference plane, first draw the simple true-shape view, then tilt and project the other view.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Draw the pentagon first in the view where its true shape is seen, then tilt its front view to \(30^\circ\) with H.P. and project the final top view.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
- Use object lines for visible final edges and thin construction lines for projectors.
Projection tip
The final view must satisfy the given angle and the stated distance from H.P. or V.P.Construction route. For lamina problems, do not start with the inclined view directly. The true-shape view is the control drawing.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Draw the pentagon first in the view where its true shape is seen, then tilt its front view to \(30^\circ\) with H.P. and project the final top view.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
Draw the projections of a triangular lamina of 30 mm sides, having side \(AB\) in V.P. and its surface inclined at \(60^\circ\) to V.P.
Concept used. A plane figure shows true shape only in the view parallel to its surface. If the plane is inclined to one reference plane, first draw the simple true-shape view, then tilt and project the other view.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Start with the true triangular shape in front view, keep \(AB\) on V.P., tilt the plane to \(60^\circ\) to V.P. in top view, and project back.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
- Use object lines for visible final edges and thin construction lines for projectors.
Drawing check
The final view must satisfy the given angle and the stated distance from H.P. or V.P.Construction route. For lamina problems, do not start with the inclined view directly. The true-shape view is the control drawing.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Start with the true triangular shape in front view, keep \(AB\) on V.P., tilt the plane to \(60^\circ\) to V.P. in top view, and project back.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
A square plate with 35 mm sides is inclined at \(45^\circ\) to V.P. and perpendicular to H.P. Draw projections if one corner is in V.P. and the two sides containing that corner are equally inclined to V.P.
Concept used. A plane figure shows true shape only in the view parallel to its surface. If the plane is inclined to one reference plane, first draw the simple true-shape view, then tilt and project the other view.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Draw the true square in top view with the selected corner on V.P., set the adjacent sides symmetrically, then project the front view.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
- Use object lines for visible final edges and thin construction lines for projectors.
Projection tip
The final view must satisfy the given angle and the stated distance from H.P. or V.P.Construction route. For lamina problems, do not start with the inclined view directly. The true-shape view is the control drawing.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Draw the true square in top view with the selected corner on V.P., set the adjacent sides symmetrically, then project the front view.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
A hexagonal plate of 30 mm sides rests on the ground on one side parallel to V.P. The surface of the lamina is inclined at \(45^\circ\) to H.P. Draw its projections.
Concept used. A plane figure shows true shape only in the view parallel to its surface. If the plane is inclined to one reference plane, first draw the simple true-shape view, then tilt and project the other view.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Draw the hexagon true shape in top view first, project front view, tilt the front view to \(45^\circ\) with H.P., and obtain the final top view.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
- Use object lines for visible final edges and thin construction lines for projectors.
Drawing check
The final view must satisfy the given angle and the stated distance from H.P. or V.P.Construction route. For lamina problems, do not start with the inclined view directly. The true-shape view is the control drawing.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Draw the hexagon true shape in top view first, project front view, tilt the front view to \(45^\circ\) with H.P., and obtain the final top view.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
A rectangular lamina measuring 25 mm \(\times\) 20 mm is parallel to and 15 mm above H.P. Draw projections when one longer edge makes \(30^\circ\) with V.P.
Concept used. A plane figure shows true shape only in the view parallel to its surface. If the plane is inclined to one reference plane, first draw the simple true-shape view, then tilt and project the other view.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Draw the true rectangle in top view at \(30^\circ\) to \(XY\), then project the front view 15 mm above \(XY\).
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
- Use object lines for visible final edges and thin construction lines for projectors.
Projection tip
The final view must satisfy the given angle and the stated distance from H.P. or V.P.Construction route. For lamina problems, do not start with the inclined view directly. The true-shape view is the control drawing.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- Draw the true rectangle in top view at \(30^\circ\) to \(XY\), then project the front view 15 mm above \(XY\).
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
Draw the projections of a circle of 30 mm diameter, having its plane vertical and inclined at \(30^\circ\) to V.P. Its centre is 25 mm above H.P. and 20 mm in front of V.P.
Concept used. A plane figure shows true shape only in the view parallel to its surface. If the plane is inclined to one reference plane, first draw the simple true-shape view, then tilt and project the other view.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- A vertical circle appears as a line in top view and a true circle in front view before inclination. Tilt the top view to \(30^\circ\) and project the final ellipse-like front view.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
- Use object lines for visible final edges and thin construction lines for projectors.
Drawing check
The final view must satisfy the given angle and the stated distance from H.P. or V.P.Construction route. For lamina problems, do not start with the inclined view directly. The true-shape view is the control drawing.
- Draw \(XY\) and choose the first view where the lamina is seen in true shape.
- A vertical circle appears as a line in top view and a true circle in front view before inclination. Tilt the top view to \(30^\circ\) and project the final ellipse-like front view.
- Project all corners to obtain the companion view.
- Redraw the inclined view at the specified angle, keeping the given edge or point condition fixed.
- Project again from the inclined view to obtain the final required view.
Project the front view and top view of a square prism of 35 mm base edges and 50 mm vertical height, resting on H.P., with two vertical rectangular faces parallel to V.P.
Concept used. When a solid has its axis perpendicular to H.P., start with the top view because the base is seen in true shape.
- Draw \(XY\).
- Draw the true-shape top view of the square prism below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape top view of the square prism below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
A triangular prism of 40 mm base edges and 60 mm height stands on H.P. with one vertical rectangular face at the rear, parallel to V.P. Draw its projections.
Concept used. When a solid has its axis perpendicular to H.P., start with the top view because the base is seen in true shape.
- Draw \(XY\).
- Draw the true-shape top view of the triangular prism below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape top view of the triangular prism below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
Draw the projections of a cylinder resting on H.P. on its base, with 30 mm base diameter and 40 mm long axis.
Concept used. When a solid has its axis perpendicular to H.P., start with the top view because the base is seen in true shape.
- Draw \(XY\).
- Draw the true-shape top view of the cylinder below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape top view of the cylinder below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
Project the front view and top view of a hemisphere resting on H.P. with its circular face on top. Diameter is 60 mm.
Concept used. When a solid has its axis perpendicular to H.P., start with the top view because the base is seen in true shape.
- Draw \(XY\).
- Draw the true-shape top view of the hemisphere below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape top view of the hemisphere below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
Project the front view and top view of the frustum of a hexagonal pyramid, 25 mm base edges and 70 mm height, cut at mid-height parallel to its base.
Concept used. When a solid has its axis perpendicular to H.P., start with the top view because the base is seen in true shape.
- Draw \(XY\).
- Draw the true-shape top view of the hexagonal pyramid frustum below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape top view of the hexagonal pyramid frustum below \(XY\) using the given base size.
- Project all base corners or generators upward to obtain the front view.
- Set the given axis or height in front view.
- Join visible edges by object lines and hidden edges by short dashed lines where needed.
Project the front view and top view of a hollow cylinder having outer diameter 50 mm, inner diameter 40 mm and length 50 mm, resting on H.P. with its axis perpendicular to V.P.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the hollow cylinder above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the hollow cylinder above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
A triangular pyramid of 50 mm base and 50 mm axis rests on its base corner on H.P., so that the upper edge of the base is horizontal. The base is at the rear and parallel to V.P. Draw projections.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the triangular pyramid above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the triangular pyramid above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Project the front view and top view of a pentagonal prism of 30 mm base edges and 60 mm long edges perpendicular to V.P., with its rectangular face on top parallel to H.P.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the pentagonal prism above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the pentagonal prism above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
A frustum of a square pyramid of 40 mm base edges and 20 mm top edges rests on H.P. on a base edge. Its 50 mm axis is horizontal and at right angles to V.P.; the cut face is in front. Draw projections.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the square pyramid frustum above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the square pyramid frustum above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
A hexagonal prism of 25 mm base edge and 60 mm axis rests on one base edge on H.P. and its axis is perpendicular to V.P. Project its front view and top view.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the hexagonal prism above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the hexagonal prism above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Project the front view and top view of a cylinder with base diameter 50 mm and height 70 mm, resting on H.P., with its axis perpendicular to V.P.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the cylinder above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the cylinder above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Draw the front view and top view of a cone of base diameter 30 mm and axis 65 mm, with its axis perpendicular to V.P., keeping the vertex in front.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the cone above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the cone above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
A square prism, base 40 mm side and axis 70 mm long, lies on one rectangular face. Its axis is perpendicular to V.P. Draw its front view and top view.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the square prism above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the square prism above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
The frustum of a triangular pyramid of 50 mm base edge and 20 mm top edge rests on H.P. with its base edge on it. The 60 mm axis is parallel to H.P. and at right angles to V.P. The cut face is in front. Project its views.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the triangular pyramid frustum above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the triangular pyramid frustum above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
A right regular pentagonal pyramid of base edge 25 mm and height 60 mm has its axis perpendicular to V.P. and its base parallel to V.P. Draw its projections.
Concept used. When a solid has its axis perpendicular to V.P., start with the front view because the end face is seen in true shape.
- Draw \(XY\).
- Draw the true-shape front view of the pentagonal pyramid above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Draw the true-shape front view of the pentagonal pyramid above \(XY\).
- Project downward from all corners, rim points or generators.
- Set the given axis length in top view.
- Complete the top view and apply hidden lines only where rear edges are not visible.
A hexagonal prism, base 25 mm side and axis 60 mm long, lies on the ground on one face with the axis parallel to both V.P. and H.P. Draw projections.
Concept used. When the axis is parallel to both H.P. and V.P., one view usually shows the axis in true length and the other view gives the contact condition.
- Draw \(XY\).
- Place the hexagonal prism in the simple lying position that satisfies the contact with H.P.
- Draw the view in which the base or end face is easiest to control.
- Project the second view using perpendicular projectors.
- Show the axis parallel to \(XY\) in both views and use dashed lines for hidden edges.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Place the hexagonal prism in the simple lying position that satisfies the contact with H.P.
- Draw the view in which the base or end face is easiest to control.
- Project the second view using perpendicular projectors.
- Show the axis parallel to \(XY\) in both views and use dashed lines for hidden edges.
A triangular pyramid, base 25 mm side and axis 50 mm long, rests on the ground on one base edge. Its axis is parallel to both planes. Draw projections.
Concept used. When the axis is parallel to both H.P. and V.P., one view usually shows the axis in true length and the other view gives the contact condition.
- Draw \(XY\).
- Place the triangular pyramid in the simple lying position that satisfies the contact with H.P.
- Draw the view in which the base or end face is easiest to control.
- Project the second view using perpendicular projectors.
- Show the axis parallel to \(XY\) in both views and use dashed lines for hidden edges.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Place the triangular pyramid in the simple lying position that satisfies the contact with H.P.
- Draw the view in which the base or end face is easiest to control.
- Project the second view using perpendicular projectors.
- Show the axis parallel to \(XY\) in both views and use dashed lines for hidden edges.
A cylinder, base 40 mm diameter and axis 60 mm long, lies on the ground on its generators with the axis parallel to both V.P. and H.P. Draw projections.
Concept used. When the axis is parallel to both H.P. and V.P., one view usually shows the axis in true length and the other view gives the contact condition.
- Draw \(XY\).
- Place the cylinder in the simple lying position that satisfies the contact with H.P.
- Draw the view in which the base or end face is easiest to control.
- Project the second view using perpendicular projectors.
- Show the axis parallel to \(XY\) in both views and use dashed lines for hidden edges.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Place the cylinder in the simple lying position that satisfies the contact with H.P.
- Draw the view in which the base or end face is easiest to control.
- Project the second view using perpendicular projectors.
- Show the axis parallel to \(XY\) in both views and use dashed lines for hidden edges.
A frustum of a hexagonal pyramid of 20 mm base edges and 10 mm top edges rests on H.P. on a base edge with its 50 mm axis horizontal and parallel to V.P. The cut top face is in front. Draw projections.
Concept used. When the axis is parallel to both H.P. and V.P., one view usually shows the axis in true length and the other view gives the contact condition.
- Draw \(XY\).
- Place the hexagonal pyramid frustum in the simple lying position that satisfies the contact with H.P.
- Draw the view in which the base or end face is easiest to control.
- Project the second view using perpendicular projectors.
- Show the axis parallel to \(XY\) in both views and use dashed lines for hidden edges.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw \(XY\).
- Place the hexagonal pyramid frustum in the simple lying position that satisfies the contact with H.P.
- Draw the view in which the base or end face is easiest to control.
- Project the second view using perpendicular projectors.
- Show the axis parallel to \(XY\) in both views and use dashed lines for hidden edges.
A triangular prism with 25 mm edges at its base and axis 60 mm long rests on one base edge with axis parallel to V.P. and inclined at \(30^\circ\) to H.P. Draw projections.
Concept used. For an axis parallel to V.P. and inclined to H.P., first assume the axis perpendicular to H.P.; then tilt the front view to the required angle.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
- Join corresponding points and mark visible and hidden edges.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
A pentagonal pyramid of 25 mm base edges and 50 mm axis has its axis perpendicular to V.P. and 50 mm above H.P. Draw projections if one base edge is inclined at \(30^\circ\) to H.P.
Concept used. For an axis parallel to V.P. and inclined to H.P., first assume the axis perpendicular to H.P.; then tilt the front view to the required angle.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
- Join corresponding points and mark visible and hidden edges.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
A frustum of square pyramid of 20 mm top edges, 40 mm bottom edges and 50 mm axis has its side face inclined at \(45^\circ\) to H.P. with axis parallel to V.P. Draw projections.
Concept used. For an axis parallel to V.P. and inclined to H.P., first assume the axis perpendicular to H.P.; then tilt the front view to the required angle.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
- Join corresponding points and mark visible and hidden edges.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
The frustum of a cone has 90 mm base diameter and 30 mm top diameter. Draw projections when its axis is parallel to V.P. and inclined at \(30^\circ\) to H.P.
Concept used. For an axis parallel to V.P. and inclined to H.P., first assume the axis perpendicular to H.P.; then tilt the front view to the required angle.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
- Join corresponding points and mark visible and hidden edges.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
The frustum of a cone has 90 mm base diameter and 30 mm top diameter. Draw projections when its axis is parallel to V.P. and inclined at \(60^\circ\) to H.P.
Concept used. For an axis parallel to V.P. and inclined to H.P., first assume the axis perpendicular to H.P.; then tilt the front view to the required angle.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
- Join corresponding points and mark visible and hidden edges.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the simple top view with the axis perpendicular to H.P.
- Project the first front view.
- Redraw the front view so that the axis makes the given angle with \(XY\).
- Project down from the inclined front view.
- Use horizontal loci from the first top view to locate the final top-view points.
Draw projections of a pentagonal prism having 25 mm base edge and 50 mm axis when it rests on its base with one base edge inclined at \(30^\circ\) to V.P.
Concept used. For an axis parallel to H.P. and inclined to V.P., first assume the axis perpendicular to V.P.; then tilt the top view to the required angle.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
- Join corresponding points and mark visible and hidden edges.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
A triangular pyramid of 50 mm base edges and 60 mm axis has one base corner touching V.P., with axis parallel to H.P. and inclined at \(45^\circ\) to V.P. Draw projections.
Concept used. For an axis parallel to H.P. and inclined to V.P., first assume the axis perpendicular to V.P.; then tilt the top view to the required angle.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
- Join corresponding points and mark visible and hidden edges.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
A frustum of a cone of 40 mm base diameter and 20 mm cut-face diameter rests on H.P. Its 50 mm axis is parallel to H.P. and inclined to V.P. at \(30^\circ\) towards right. Project top view and front view.
Concept used. For an axis parallel to H.P. and inclined to V.P., first assume the axis perpendicular to V.P.; then tilt the top view to the required angle.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
- Join corresponding points and mark visible and hidden edges.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
A square duct is a frustum of a square pyramid. The top and bottom sides are 90 mm and 60 mm, and the length is 110 mm. Its axis is parallel to H.P. and inclined at \(60^\circ\) to V.P. Draw projections, neglecting sheet thickness.
Concept used. For an axis parallel to H.P. and inclined to V.P., first assume the axis perpendicular to V.P.; then tilt the top view to the required angle.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
- Join corresponding points and mark visible and hidden edges.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
Draw projections of a square prism having 30 mm base edge and 55 mm axis when it rests on its base with its axis inclined at \(30^\circ\) to V.P.
Concept used. For an axis parallel to H.P. and inclined to V.P., first assume the axis perpendicular to V.P.; then tilt the top view to the required angle.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
- Join corresponding points and mark visible and hidden edges.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
Draw projections of a hexagonal prism having 20 mm base edge and 50 mm axis when it rests on its base, with axis parallel to H.P. and inclined at \(40^\circ\) to V.P.
Concept used. For an axis parallel to H.P. and inclined to V.P., first assume the axis perpendicular to V.P.; then tilt the top view to the required angle.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
- Join corresponding points and mark visible and hidden edges.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
A triangular prism of 50 mm base edges and 60 mm axis rests on its base and has its axis inclined at \(45^\circ\) to V.P. Draw projections.
Concept used. For an axis parallel to H.P. and inclined to V.P., first assume the axis perpendicular to V.P.; then tilt the top view to the required angle.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
- Join corresponding points and mark visible and hidden edges.
Projection tip
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
A hexagonal pyramid of 25 mm base edges and 60 mm axis rests on its base and has its axis inclined to V.P. at \(30^\circ\). Draw projections.
Concept used. For an axis parallel to H.P. and inclined to V.P., first assume the axis perpendicular to V.P.; then tilt the top view to the required angle.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
- Join corresponding points and mark visible and hidden edges.
Drawing check
The final axis direction and resting/contact condition must both match the question.Construction route. Keep the first simple position light. The final redrawn view is the answer view.
- Draw the true-shape front view for the simple position.
- Project the first top view.
- Redraw the top view so that the axis makes the given angle with \(XY\).
- Project upward from the inclined top view.
- Use horizontal loci from the first front view to locate the final front-view points.
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