08-03-2025 | General Studies | ESE

Key Concepts

• Auxiliary Inclined Plane (AIP): Plane inclined to HP and perpendicular to VP.
• Auxiliary Vertical Plane (AVP): Plane inclined to VP and perpendicular to HP.
• Oblique Plane: Plane inclined to both VP and HP.
• Vertical Trace (VT): Intersection of a plane with the VP.
• Horizontal Trace (HT): Intersection of a plane with the HP.
• True Shape and Size: Actual shape and size of a plane as seen when viewed perpendicularly.
• Polyhedra: Solids bounded by plane surfaces.
• Regular Polyhedra: Polyhedra with faces that are regular polygons (e.g., squares, equilateral triangles).
• Prism: Solid with two equal, parallel bases connected by rectangular faces.
• Pyramid: Solid with a base and an apex connected by isosceles triangles.
• Solid of Revolution: Solid formed by rotating a plane figure around an axis.
• Sectioning of Solids: Process of cutting a solid to reveal its internal structure.
• Hatching Lines: Lines used to indicate the cut surface in a sectioned view.

Projection of Planes: Inclined and Oblique Planes

Auxiliary Inclined Plane (AIP)

• Definition: A plane inclined to the Horizontal Plane (HP) and perpendicular to the Vertical Plane (VP).
• Example: A square ABCD inclined at 30 degrees to HP and perpendicular to VP, with side AB parallel to HP.
• Front View: A line inclined at 30 degrees to the XY line, showing C'D' and A'B' (A' hidden).
• Top View: A distorted view of the square, showing A, B, C, and D.
• Traces:
  • Vertical Trace (VT): The entire line in the front view.
  • Horizontal Trace (HT): A line from the top view where the plane intersects the HP.
• True Shape and Size: Not visible in either the front or top view. To obtain the true shape and size, an auxiliary top view is needed, where the observer looks perpendicularly to the inclined plane.
• Key Points:
  • The front view of an AIP gives the vertical trace.
  • The horizontal trace is a line perpendicular to the XY line.
  • The view obtained on the AIP is known as the auxiliary top view.

Auxiliary Vertical Plane (AVP)

• Definition: A plane inclined to the Vertical Plane (VP) and perpendicular to the Horizontal Plane (HP).
• Example: A square ABCD inclined at 45 degrees to VP and perpendicular to HP, with side AB parallel to HP.
• Front View: A distorted view of the square, showing A', B', C', and D'.
• Top View: A line inclined at 45 degrees to the XY line, showing C, D, and A, B (A hidden).
• Traces:
  • Horizontal Trace (HT): The entire line in the top view.
  • Vertical Trace (VT): A line perpendicular to the XY line.
• True Shape and Size: Not visible in either the front or top view. To obtain the true shape and size, an auxiliary front view is needed, where the observer looks perpendicularly to the inclined plane.
• Key Points:
  • The top view of an AVP gives the horizontal trace.
  • The vertical trace is a line perpendicular to the XY line.
  • The view obtained on the AVP is known as the auxiliary front view.

Oblique Plane

• Definition: A plane inclined to both the VP and HP.
• Case 1: θ + φ ≠ 90°
  • Construction Stages: Requires three stages to draw the multi-view projection.
    1. Plane parallel to HP and perpendicular to VP.
    2. Plane inclined to HP and perpendicular to VP (AIP).
    3. Plane inclined to both VP and HP (Oblique Plane).
• Case 2: θ + φ = 90°
  • The plane is placed such that the edge view or line view is seen in the profile plane.
  • The true length of the axis is visible in the profile plane.
  • The front and top views of the axis are perpendicular to the XY line.

Projection of Solids

Types of Solids

1. Polyhedra: Solids bounded by plane surfaces.
   • Regular Polyhedra: Solids bounded by regular polygons.
     • Tetrahedron: Four equilateral triangular faces.
     • Hexahedron (Cube): Six square faces.
     • Octahedron: Eight equilateral triangular faces.
     • Dodecahedron: Twelve regular pentagonal faces.
     • Icosahedron: Twenty equilateral triangular faces.
2. Prisms: Two equal-sized, parallel bases connected by rectangular faces.
   • The line passing through the midpoints of the bases is called the axis.
   • Named based on the shape of the base (e.g., square prism, triangular prism).
3. Pyramids: A base and an apex connected by isosceles triangles.
   • The line passing through the apex and the center of the base is called the axis.
   • Named based on the shape of the base (e.g., square pyramid, triangular pyramid).
4. Solids of Revolution: Solids formed by rotating a plane figure around an axis.
   • Sphere: Formed by rotating a semicircle.
   • Cylinder: Formed by rotating a rectangle.
   • Cone: Formed by rotating a triangle.

Cases of Solid Orientation

1. Axis Perpendicular to HP and Parallel to VP: The front view shows the true shape of the base, and the top view is a line.
2. Axis Perpendicular to VP and Parallel to HP: The top view shows the true shape of the base, and the front view is a line.
3. Axis Inclined to One Reference Plane and Parallel to the Other: Requires multiple views to determine the true shape and size.
4. Axis Inclined to Both VP and HP:
   • θ + φ ≠ 90°: Alpha > Theta and Beta > Phi
   • θ + φ = 90°: True length of axis in profile plane, front and top views of axis are perpendicular to XY.

Example Problem: Tetrahedron

• Problem: A tetrahedron resting in HP on its face, with one side perpendicular to VP. Determine the front and top views.
• Solution:
  • Front View: An isosceles triangle.
  • Top View: An equilateral triangle.

Example Problem: Cube

• Problem: A cube which is equally inclined to VP and resting on its face in HP has solid diagonal perpendicular to HP. What is the top view?
• Solution: A regular hexagon.

Section of Solids

Purpose

• Sectioning is done to reveal the internal structure of a solid, which is shown by hatching lines.

Section of Tetrahedron

• Example: A tetrahedron resting in HP with one side perpendicular to VP.
• Maximum Point of Intersections: 4
• General Formula for Pyramids: N+1, where N is the number of sides in the base.

Section of Hexahedron (Cube)

• Maximum Point of Intersections: 6
• General Formula for Prisms: N+2, where N is the number of sides in any one base.

Conclusion

The video provides a detailed explanation of the projection of planes and solids, including inclined and oblique planes, different types of solids, and sectioning techniques. It emphasizes the importance of understanding the orientation of the object with respect to the reference planes and the use of auxiliary views to obtain the true shape and size. The video also provides practical examples and formulas to help solve problems related to sectioning of solids.