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Lesson Ten

Learn the fundamental principle and steps of the cutting-plane method, a powerful way to find intersections between revolutionary surfaces.

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Lesson Ten

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  1. Lesson Ten Ⅰ.Method 2— cutting-plane method (截平面法) 1.Fundamental principle The cutting-plane method is a powerful, efficient method for finding intersections. Suppose that a cutting plane passes through each the given revolutionary surfaces simultaneously, the cutting plane will intersect with two the given revolutionary surfaces and produce intersections respectively. The intersecting points intersected by these intersections are common points and they belongs to three planes (三面共点). Namely a common point belongs to the cutting plane and two revolutionary surfaces.

  2. Look at picture, the intersecting point Ⅰbelongs to conical surface (on a latitude circle), cylindrical surface (on a element) and cutting plane P. Connecting the view of a series of common points on the same projection plane, we will obtain the intersection for both revolutionary surfaces. Ⅰ

  3. 2. Select cutting plane: 1) Cutting plane should be in intersecting range. 2) When cutting plane intersects with two revolutionary surfaces, producing intersection should be simple intersection (straight line or circle).

  4. For example 1: Complete three views

  5. Analyse: Intersecting two revolutionary surfaces are partial sphere surface and conical surface. Intersection is a closed spatial curve. Method 1 is invalid as there is no cylindrical surface in the solid. We only make use of method 2 to solve the problem.

  6. Solution: 1) Peculiar points: ① Points on the frontal outlines Suppose that the solid is cut by a cutting plane P which passes through apex and is parallel to V plane, the cutting plane P will intersect with conical surface and sphere surface. And the former intersections are front outlines of the cone and the latter intersection is front outline of the sphere. Their intersecting points, point A and point B, are a pair of common points. Look at picture please.

  7. Making use of point A, B on the frontal outlines of the cone, we obtain top view, a and b, and left view, (a″) and b″.

  8. ②Points on the profile outlines. Suppose that the solid is cut by a cutting plane Q which passes through apex and is parallel to W plane, the cutting plane Q will intersect with conical surface and sphere surface. And the former intersections are profile outlines of the cone and the latter intersection is profile latitude circle of the sphere. Their intersecting points, point C and point D, are a pair of common points. Look at picture please.

  9. Making use of point C, D on the profile outlines of the cone, we obtain top view, c and d, and front view, c and (d).

  10. 2) General points Suppose that the solid is cut by a cutting plane R which is perpendicular to the axis of the cone and parallel to H plane, the cutting plane R will intersect with conical surface and sphere surface. And the former intersection is a horizontal latitude circle of the cone and the latter intersection is a horizontal latitude circle of the sphere. Their intersecting points, point Ⅰ and point Ⅱ, are a pair of common points. Look at picture please.

  11. Making use of point Ⅰ, Ⅱ on the cutting plane R, we obtain front view, 1 and (2), and left view, 1″ and 2 ″.

  12. 3) Connect the same views of all point and invisible is indicated by dashed line.

  13. 4) Deepen reserved outlines Finish drawing.

  14. Ⅱ. Many- solid intersect (多体相贯) Many- solids intersecting are actually both revolutionary surfaces intersecting respectively. 多体相贯实际上是两两相贯。

  15. For example 2: Complete front view and left view.

  16. Analyse: The solid is consisted by three cylinders of which two cylinders (put horizontally) share one axis and is perpendicular tothe axis of another cylinder (put vertically) There is a intersection between the cylinder put vertically and the bigger cylinder put horizontally. Top view of the intersection locates on the circumference and left view locates partial bigger circumference between two outlines.

  17. There are two intersections (two elements) between the cylinder put vertically and profile side of locating between two cylinder. Top view of the intersection locates on the Circumference (two convergence points). Profile side

  18. There is a intersection between the cylinder put vertically and the smaller cylinder put horizontally. Top view of the intersection locates on the circumference and left view locates smaller circumference.

  19. Solution: 1) Complete front view of intersection ECADG. First peculiar points on the outlines.

  20. Secondly general points on the cylindrical surface. Point E, G locate cylindrical surface and profile side.

  21. 2) Complete front view and left view of intersection EF, GH .

  22. Complete front view of intersection HBF.

  23. 4) Deepen reserved outlines and convergence lines. Finish drawing.

  24. For example 3: Complete front view.

  25. Analyse: The main part of the solid is cylinder which is put Horizontally and the axis is perpendicular to W plane. There is a boss on the upper of the cylinder. There is a hole on the boss.

  26. Solution: 1) Supplement front view of full solid.

  27. 2) Find out the given views of left external surface intersection. And we know they are equal diameter intersection.

  28. Draw front view of the intersection.

  29. 3) Supplement front view of right profile side on the boss. Because frontal side is tangent to cylindrical surface, there is no boundary in front view.

  30. 4) Supplement internal intersections. 5) Deepen reserved outlines.

  31. For example 4: Complete front view and top view.

  32. Solution:

  33. For example 5: Complete front view.

  34. Solution:

  35. For example 6: Complete top view.

  36. Solution:

  37. For example 7: Analyse intersections on the solid surface.

  38. For example 8: Complete front view and top view.

  39. For example 9: Complete front view and top view.

  40. Because a cylinder and a cone are tangent to a sphere, the intersection is changed into plane curve, ellipse. Left view of two ellipses locate circumference and front view are two convergence lines and top views are two similar shape.

  41. Solution:

  42. For example 10: Complete front view and top view.

  43. Analyse: in the problem, a cylinder intersects with a cone and produce two intersections which locate upper and down respectively. The given view of the intersection is left view which are two partial circumference between outlines.

  44. Solution:

  45. For example 11: Analyse intersections on the solid surface. There are two

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