Sectional Views of Solids: Sectional Plane Inclined to HP

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1. Orientation: Auxiliary Inclined Plane (AIP)

When a cutting plane is inclined to the Horizontal Plane (HP) at angle $\theta$ and perpendicular to the Vertical Plane (VP):

• In Front View (on VP): The cutting plane appears as a single straight line inclined at angle $\theta$ to $XY$ (Vertical Trace, $VT$).
• In Top View (on HP): The cut surface appears as the Sectional Top View, but its length along the incline is foreshortened (compressed).
• True Shape of Section: Can only be observed by looking strictly perpendicular to the cutting plane in the Front View. It must be projected onto an auxiliary plane parallel to the cutting plane line.

                   FRONT VIEW (Elevation)
                         /\
                        /  \  Cutting Plane S-S (at angle theta to XY)
                   ----+----+---- S
                      / \  / \
                  ---+---+----+--- XY (Base on HP)
                     |   |    |
                   SECTIONAL TOP VIEW (Foreshortened)
                     +--------+
                     |////////|  (Apparent section, hatched)
                     +--------+

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2. Step-by-Step Construction Procedure

Step 1: Draw Normal Projections

• Draw the standard Top View and Front View of the solid with light construction lines ($3\text{H}/4\text{H}$).

Step 2: Draw the Cutting Plane in the Front View

• Draw the cutting plane line $S\text{---}S$ inclined at the given angle $\theta$ to the $XY$ line, passing through the specified height (e.g., mid-height of the axis).
• Mark thick ends and directional arrows pointing downwards (towards the base).
• Label the cutting points where $S\text{---}S$ intersects the lateral edges and base as $1', 2', 3', \dots, n'$.

Step 3: Construct the Sectional Top View

• From each cut point $1', 2', 3', \dots$ in the Front View, project lines vertically downward to intersect the corresponding edges in the Top View to locate $1, 2, 3, \dots, n$.
• Special Case (Projector along a vertical edge or axis):
  • If a cutting point falls on an edge that is vertical in the Front View (e.g., the axis of a cylinder or a pyramid edge perpendicular to $XY$), its top view cannot be located by a simple vertical drop.
  • Fix: Draw a horizontal line from the point in the Front View to an extreme generator/edge, drop vertically down to the centerline in the Top View, and swing an arc around the center back to the original edge.
• Connect points $1, 2, 3, \dots$ sequentially (straight lines for prisms/pyramids, smooth curve for cylinders/cones).
• Hatch the enclosed area at $45^\circ$ with thin Type B lines.

Step 4: Construct the True Shape of the Section

1. Draw an auxiliary reference line $X_1Y_1$ parallel to the cutting plane line $S\text{---}S$ in the Front View at any convenient distance.
2. From each cutting point $1', 2', 3', \dots$ on $S\text{---}S$ in the Front View, project lines strictly perpendicular to $X_1Y_1$.
3. Measure the perpendicular distance of each corresponding point $1, 2, 3, \dots$ from the $XY$ line in the Top View.
4. Transfer these distances along the perpendicular projectors from $X_1Y_1$:

5. Connect points $1_T, 2_T, 3_T, \dots$ with a smooth curve or straight lines.
6. Hatch the profile at $45^\circ$ to $X_1Y_1$. This is the TRUE SHAPE OF THE SECTION.

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3. Conic Sections Produced by Cutting a Right Circular Cone

When an inclined cutting plane slices a right circular cone (semi-apex angle $= \alpha$):

                     /\  Apex
                    /  \
                   /    \   Theta > Alpha  ===> Ellipse
                  /======\  Theta = 0      ===> Circle
                 /      / \ Theta = Alpha  ===> Parabola
                /      /   \
               /      /     \
              +------+-------+ Base
                     |
                     +---> Theta < Alpha (or vertical) ===> Hyperbola

Angle of Cutting Plane ($\theta$) with Axis/Base	Position Relative to Generator	Resulting Conic (True Shape)
$\theta = 0^\circ$ to base ($\perp$ to axis)	Parallel to base	Circle
$\theta > \alpha$ to axis (cuts all generators)	Slices both sides of cone completely	Ellipse
$\theta = \alpha$ to axis (parallel to generator)	Parallel to one extreme generator	Parabola
$\theta < \alpha$ to axis (or parallel to axis)	Slices through base and one side	Hyperbola
Passing through Apex	Through vertex $O$	Pair of intersecting straight lines (Triangle)

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4. Summary of Key Exam Points

1. Cutting Plane Representation in FV: Appears as an inclined line $S\text{---}S$ making angle $\theta$ with $XY$.
2. True Shape Projection Direction: Perpendicular to $S\text{---}S$ in the Front View.
3. Coordinate Transfer Rule for AIP:

4. Cylinder Cut by Inclined Plane: The true shape of the section of a cylinder cut by an inclined plane cutting all generators is always an Ellipse (Major axis $= \text{length on } S\text{---}S$, Minor axis $= \text{cylinder diameter } D$).
5. Hatching Rule: Hatching lines are drawn at $45^\circ$ to the $X_1Y_1$ line.