1. A simple algorithm for designing developable Bézier surfaces CAGD(2003) 601-619 Günter Aumann Reporter: Ruikun Zhang Sep. 29,2005

2. Previous work 1. Proving (nonlinear)characterizing equations for free form surfaces to be developable. (Aumann, 1991a, 1991b; Lang and Röschel, 1992; Chalfant and Maekawa, 1998; Maekawa, 1998). 2. Using methods of projective geometry. Pottmann and Farin (1995). 3.Basing on the algorithm of de Casteljau. Chu and Séquin (2002).

3. Disadvantages Most of the known algorithms have one or more of the following restrictions. The characterizing equations can only be solved for boundary curves of low degree. Only planar boundary curves are permitted. It is difficult to control singular points. In this paper,there will be a new algorithm which avoids these disadvantages.

4. Fundamentals A ruled Bézier patch: Φ : x(u,w):

5. The straight lines X(u, 0)X(u, 1) are called the rulings r(u) of Φ. A point X ∈ Φ can be computed by the algorithm of de Casteljau. First,we compute X ∈ b.

6. Where, In the same way we get the point X ∈ c,then we compute:

7. According to the necessary and sufficient condition,we can conclude as follows: :=

8. The algorithm 1. Fixing the first control point of and ensure that Then,we get So, (1) backback

9. So for λ = μ = 1 quadrangle is a parallelogram. 2. Finding an affine transformation with Now we choose in a way that maps quadrangle onto quadrangle We get

10. If is fixed by, we find in the same way an affine transformation which maps quadrangle onto quadrangle if There is an affine transformation which maps quadrangle onto quadrangle if (2) back

11. For the affine transformation Defining So,

12. i.e., Theorem 1. If we choose the control points of the boundary curve c as above, for arbitrary λ,μ ∈ R the resulting Bézier surface x(u,w) is a developable surface.

14. Discussion of the presented method From (2) we get(2) (3) For λ = 1 Eq. (3) becomes from (1) we get(1) Then,

15. So from the algorithm of de Casteljau we get Theorem 2. The developable Bézier surface Φ of Theorem 1 is a cylinder for all boundary curves b iff λ = 1.

16. The algorithm of de Casteljau.

17. Examples

18. Let λ 1. For μ = 1 Eq. (3) becomes (1)shows Finally,we get Because of the algorithm of de Casteljau gives

19. So we have Theorem 3. The developable Bézier surface Φ of Theorem 1 is a cone for all boundary curves b iff μ = 1 and λ 1.

20. Examples of cones

21. Examples

22. For μ = 0 Eq. (3) becomes For λ 0, is a singular point of Φ, λ = 1,, according to Theorem 2 this gives a cylinder. The lengths of its rulings converge against zero.

23. Examples of cylinders with μ = 0.

24. About avoiding singular points

25. Let In the coordinate system We get And in the same half plane

26. (4) For λ = 1 condition (4) is true. For ρ = 0 from (4) we get λ> 0. For μ = 1 function has two distinct zeros

27. For μ> 1,. For μ< 1 we have and Theorem 4. The developable Bézier patch of Theorem 1 does not contain singular points iff λ = 1(cylindrical case) or if the starting quadrangle is convex.

28. Examples of singular points

30. Vanishing ruling.

31. Interpolation Question: Given are points P,Q on a line and points R,S on a line. Wanted is a developable surface (1) with or better

32. Three of these four requests can be met by choosing: It remains the question, how we can get or even We distinguish between three cases. 1), we design a cylinder and choose λ = 1(th2),from(3) we get(3) or

33. So, 2) If and intersect in a point T, we design a cone. According to Theorem 3 we choose μ = 1.

34. Interpolating cones.

35. 3) the case of skew lines Choosing and which gives We are searching for and with both and. Writing (3) as

36. And (4)backback Let

37. If A(μ) is a point of the plane: Let We get

38. From (4) we get(4) And

39. Let we get = 1, i.e., Let We have, i.e.,,

40. Example 1of Skew interpolation conditions

41. Example 2 of Skew interpolation conditions