Intersection of Solids

This is a common exam question and can be quite satisfying to do. It has appeared as both a short and long question. 
A question is generally broken up into 4 main parts:
  1. Reproduce the given plan and elevation (18-20 marks). Standard orthographic drawing, sometimes needs a true shape.
  2. Interpenetration of the easy side (4-6 marks). Usually just project between plan and elevation.
  3. Interpenetration of the hard side (10-15 marks). Project across surfaces, often needs an auxiliary view. 
  4. Connecting all the dots (5-12 marks). Is the edge hidden or visible? Label your points to help see what connects to what.
This means that with some decent orthographic skills and a half decent ability to visualise the object, you can about 2/3 of the marks available. The final 1/3 takes a bit more effort and here are the skills you'll need.

Terminology

Apex - the pointy tip of a pyramid or cone
Bend Points - when a line of interpenetration meets the edge of one of the shapes, it turns. The point on the edge where this happens is bend point.

Intersection - one solid going through another
Line of Interpenetration - the edge where the solids meet, this is what we're trying to find
Orthographic Views - Elevation, Plan or End View

Typical Exam Question

Always label your points - it makes things much easier!
The example below is a worked example from Graphics in Design & Communication, p186

1 - Draw Plan and Elevation

  • Let the question guide you rather than worry about where to start,  do part (a) first etc.
  • Make sure you're clear about how to go from a true shape in elevation to the plan. 
  • Label your points
Picture
True Shape
Picture
Plan

2 - Auxiliary View or Section Planes?

You need to decide whether you're going to use section planes or an auxiliary view. It can often be done with both but auxiliary works well if there's a prism at an angle. The steps below are for an auxiliary view.
  • Project along the prism (usually in elevation)
  • Take heights from XY to object in plan
  • Label all points

3 - Find Corners

  • Locate any corners of prism using Radial Elements method
  • Example to find corner 1.
    • Draw a line from B through 1 to find point X on line CD
    • Project X back to the elevation and find it on line CD
    • Connect X back to B - 1 must be on this line
    • Find where XB crosses edge 1-1 and you have point 1

4 - Find Bend Points

  • Locate any bend points, label on auxiliary and then project back to elevation. Locate by finding where the line of projection crosses the edge that the point is on. In this example 
  • In this example, points q and s are projected back to line BC in elevation and then down to plan.

5 - Plan

  • There are often several points that are quite obvious in plan view
  • Find these and project back to the elevation

6 - Completion

  • Write out the sequence of the corners and bend points, then follow that sequence to complete the lines of interpenetration, including hidden lines. 
  • It can be helpful to use the plan to figure out what's a hidden line

Method 1 - Limits

Good for:

Simpler objects with flat (plane) surfaces. 

The idea:
All interpenetration lines are straight, so by finding start (ie, 1), end (ie, 4) and bend points (ie, s) we can find the lines.

Using it:
  • Label all start, end and bend points.
  • Project between views to find their locations in each view.
  • Connect the dots in the correct order
  • Decide if a line is hidden or visible

3D Models for Worked Examples

p275 (red book), p177 (orange book)
Solidworks
eDrawings
p178 (orange book)
Solidworks
eDrawings
p179, (orange book)
Solidworks
eDrawings

Method 2 - Radial Elements

Good for:

Cones & pyramids

The idea:
Use a line from the apex (point 0) to the base (point 1) to find sample points (p & q)  on the line of interpenetration

Using it:
  • Draw a radial element (0,1) in one of the orthographic views.
  • Find the points on the line of interpenetration (ie, p & q) and project them to the other views.
  • Repeat with more radial elements until you have enough points to draw the lines of interpenetration

3D Models of Worked Examples & Questions

Picture
p188, Q4
Solidworks
eDrawings

Method 3 - Horizontal or Vertical Section Planes

These methods are listed separately in your textbook but the idea is the same with both so they are considered together here.

Good for:

Spheres, cones or cylinders

The idea:
Take a series of horizontal or vertical section planes to find points on the lines of interpenetration. The section planes will produce simple shapes such as solids or rectangles for these objects.

Tips:
If there's a sphere, take section planes equidistant to each side of the centre line, this reduces the amount of circles you need to draw.

3D Models of Worked Examples

Tip: Use the Section Plane feature in Solidworks to quickly see what the different section planes look like.
Picture
p281 (red book),  (orange book)
Solidworks
eDrawings

Method 4 - Auxiliary Plans

This method is the most common in exam questions so make sure you know it!

Good for:
Complex planar shapes where you cannot find the bend points easily

The idea:
Use an auxiliary plan to see the bend points easily, then project back to complete the drawing. 

Using it:
  • Draw an auxiliary plan along the prism's axis, ie, so that the triangular prism on the right looks like a triangle.
  • Identify the bend points
  • Project the bend points back to the elevation.
  • Use limits or radial elements ideas to find difficult bend points.

3D Models of Worked Examples & Questions
p186 (orange book)
Solidworks
eDrawings
p190, q10 (orange book)
Solidworks
eDrawings