Solids Interpenetration

Question 1

What is a curve of interpenetration ?

Answer 1

When two 3D bodies overlap or one passes through another, the surface where they meet forms a 3D curve.
That curve is the curve of interpenetration. (相贯线)
I can be a line.

Question 2

What are the characteristics of an interpenetrating line ?

Answer 2

• It is the line that belongs to both solids meeting surfaces .
• Most of time , the line is a closed curve.

Question 3

What are the methods top find the Interpenetrating line ?

Answer 3

A - Find a point or a line on the solid surface.
1. Choose elements on one solid
• For cylinder: generators (直母线)
• For cone: generators
• For sphere: circles of latitude
• For prism: edges, faces
2. Find where each element intersects the other solid
• Solve geometrically: whether the generator penetrates the second solid.
• The intersection point gives one point of the 相贯线.
3. Project these points into FV, TV, WV
Each point gives 3 projection points.
4. Connect smooth points → 相贯线
• Usually smooth curve (for curved surfaces)
• Or polygonal chain (if plane surfaces).

When to use
• Good for cylinder ↔ cylinder, cylinder ↔ prism, cone ↔ prism
• When solids have simple regular lines you can sample.
- APM
You cut both solids with a series of auxiliary planes (辅助平面).
Each plane produces a pair of section curves, and their intersections give points on the final 相贯线.

1.	Choose a family of cutting planes Choose planes that simplify the intersections:
•	Vertical auxiliary planes (辅助竖直平面)
•	Horizontal auxiliary planes (辅助水平面)
•	Inclined auxiliary planes (辅助斜面)
2.	Each auxiliary plane cuts both solids
•	Cylinder → gives a line or ellipse
•	Cone → gives a line or conic
•	Sphere → gives a circle
•	Prism → gives lines
3.	Find intersection of those section curves The intersection points are points on the 相贯线.
4.	Project these points into all views FV, TV, WV → connect smooth curve.

Question 4

Case 1 : Two cylinders interpenetrating. (Their axes are perpendicular).
The Vertical cylinder is greater than the Horizontal one.
What are the steps ?

Answer 4

Let’s name the cylinders.
A the Vertical one and B the horizontal one.
Since A<B every generators on A will meet B surface
But some generators on B will not meet A.

1-. Find where each element intersects the other solid
• Solve geometrically: whether the generator penetrates the second solid. (Smaller penetrates bigger)
• Find special intersection points of the 相贯线. (From top view , the left-most point , the rightest point , the most forward point and the most backward point )
• Find random intersection points of the 相贯线 on TV. From there , Draw vertical lines onto FV. Measure distance between axis and points to find them on WV. Then from there draw horizontal lines.
3. Project these points into FV, TV, WV
Each point gives 3 projection points.
4. Connect smooth points → 相贯线
• Usually smooth curve (for curved surfaces)
• Or polygonal chain (if plane surfaces).

Question 5

Case of a hole in a cylinder.

Answer 5

1. Imagine a cylinder filling that hole
   2.Solve it like it was there.
   3.Remove cylinder.
2. Don’t forget that this time there will be a visible interpenetrating line and another one.

Question 6

Let’s take the case of a cylinder and a cone interpenetrating.

Answer 6

1- Check from the views which one is bigger so that you could know the shape of the curve
2- Find special points(points on both cylinder and cone) , label them and find them in each view by using APM
3- Choose random points and find them.

Question 7

What about a hole in a cone case

Answer 7

It is the same as a cylinder penetrating
Use the same method.

Question 8

Make a generalized solving method of two interpenetrating revolution solids.

Answer 8

1- From projections , tell with types of revolution solids you have (cone , cylinder , sphere) and their positions from the projection views.
2-Find the the faces where solids meet
3-Find the the line of interpenetration from those faces and find the limits of the curve and its visibility
4-Draw (delete) missing part with thick or dashed line

Question 9

What are the special cases of an interpenetrating line ?

Answer 9

1- The line is a curved plane : When two coaxial solid of revolution are interpenetrating each other , the interpenetrating line must be a circle perpendicular to the common axis. And when that axis is parallel to a projection surface , the projection of this circle on that projection surface is a line perpendicular to the axis. Condition : TWO COAXIAL SOLIDS.

2-When two quadric surfaces (like cylinders, cones, spheres, paraboloids) both touch the same sphere tangentially(at one point ), the intersection curve between them is an ellipse. That ellipse lies in a plane that is perpendicular to the plane determined by the two axes of the quadric surfaces.

3- When Two same revolution solids with parallels axis touch , the interpenetrating line is a straight line.

Question 10

What are the factors that can influence the shape of the interpenetrating curve ?

Answer 10

1- Shape of solids (cylinder-cylinder is different from cone-cylinder)
2- Size of solids (the curve shape warps around the shape of the bigger solid curvature)
3-Mutual positions of solids