Curves, Surfaces and Recursive Subdivision

Transcription

1 Department of Computer Sciences Graphics Fall 25 (Lecture ) Curves, Surfaces and Recursive Subdivision Conics: Curves and Quadrics: Surfaces Implicit form arametric form Rational Bézier Forms Recursive Subdivision of Curves Recursive Subdivision of Surfaces The University of Texas at Austin

2 Department of Computer Sciences Graphics Fall 25 (Lecture ) Conic Curves Conic Sections (Implicit form) Ellipse Hyperbola arabola x 2 a 2 + y2 b 2 = a, b > x 2 a 2 y2 b 2 = a, b > y 2 = 4ax a > The University of Texas at Austin 2

3 Department of Computer Sciences Graphics Fall 25 (Lecture ) Conic Sections (arametric form) Ellipse Hyperbola arabola x(t) = a t2 + t 2 y(t) = b 2t ( < t < + ) + t2 x(t) = a + t2 t 2 y(t) = b 2t ( < t < + ) t2 The University of Texas at Austin 3

4 Department of Computer Sciences Graphics Fall 25 (Lecture ) x(t) = at 2 y(t) = 2at ( < t < + ) The University of Texas at Austin 4

5 Department of Computer Sciences Graphics Fall 25 (Lecture ) Constructing Curve Segments Linear blend: Line segment from an affine combination of points (t) = ( t) + t t ( _ t ) The University of Texas at Austin 5

6 Department of Computer Sciences Graphics Fall 25 (Lecture ) Quadratic blend: Quadratic segment from an affine combination of line segments (t) = ( t) + t (t) = ( t) + t 2 2 (t) = ( t) (t) + t (t) 2 2 The University of Texas at Austin 6

7 Department of Computer Sciences Graphics Fall 25 (Lecture ) Cubic blend: Cubic segment from an affine combination of quadratic segments (t) = ( t) + t (t) = ( t) + t 2 2 (t) = ( t) (t) + t (t) 2 (t) = ( t) 2 + t 3 2 (t) = ( t) (t) + t 2 (t) 3 (t) = ( t) 2 (t) + t 2 (t) The University of Texas at Austin 7

8 Department of Computer Sciences Graphics Fall 25 (Lecture ) The pattern should be evident for higher degrees 3 The University of Texas at Austin 8

9 Department of Computer Sciences Graphics Fall 25 (Lecture ) Geometric view (de Casteljau Algorithm): Join the points i by line segments Join the t : ( t) points of those line segments by line segments Repeat as necessary The t : ( t) point on the final line segment is a point on the curve The final line segment is tangent to the curve at t t ( _ ) t _ ) ( t 2 t ( _ t) t 2 The University of Texas at Austin 9

10 Department of Computer Sciences Graphics Fall 25 (Lecture ) Subdivision of olygons Four oint Scheme c j -2 j+ c-3 j+ c-2 j - c j+ j+ c - c j+ c j c c 2 j c j+ j+ c 3 c j -3 c j 2 c j 3 Four point scheme: the filled circles are the level j control points, the filled squares are the level j + control points. For four-point scheme we need to consider only 7 control points; these 7 points completely define the piece of the curve around a control point. We can consider a set of 7 control points on any subdivision level, as we do not care how small our piece of the curve is. Note that we can compute the positions of the seven control points on level j + from the positions of similar seven control points on level j, using a 7 7 submatrix S of the infinite subdivision matrix. The University of Texas at Austin

11 Department of Computer Sciences Graphics Fall 25 (Lecture ) The local subdivision matrix for the four-point scheme is: c j+ 3 c j+ 2 c j+ c j+ c j+ c j+ 2 c j+ 3 = c j 3 c j 2 c j c j c j c j 2 c j 3 The University of Texas at Austin

12 Department of Computer Sciences Graphics Fall 25 (Lecture ) Quadric Surfaces Implicit form arametric form The University of Texas at Austin 2

13 Department of Computer Sciences Graphics Fall 25 (Lecture ) Constructing Surface atches Triangular decasteljau: Join adjacently indexed ijk by triangles Find r : s : t barycentric point in each triangle Join adjacent points by triangles Repeat Final point is the surface point (r, s, t) final triangle is tangent to the surface at (r, s, t) Triangle up/down schemes become tetrahedral up/down schemes roperties: The University of Texas at Austin 3

14 Department of Computer Sciences Graphics Fall 25 (Lecture ) Each boundary curve is a Bézier curve atches will be joined smoothly if pairs of boundary triangles are planar as shown Q 2 2 Q Q 2 The University of Texas at Austin 4

15 Department of Computer Sciences Graphics Fall 25 (Lecture ) Tensor roduct atches Tensor roduct atches: The control polygon is the polygonal mesh with vertices i,j The patch basis functions are products of curve basis functions (s, t) = n n i,j B n i,j (s, t) i= j= where B n i,j (s, t) = Bn i (s)bn j (t) The University of Texas at Austin 5

16 Department of Computer Sciences Graphics Fall 25 (Lecture ) The University of Texas at Austin 6

17 Department of Computer Sciences Graphics Fall 25 (Lecture ) Smoothly Joined atches: 2 3 Q 2 Q 3 Q Q 2 Q 2 Q Q 3 33 Q 3 32 Can be achieved by ensuring that ( i,n i,n ) = β(q i, Qi, ) for β > (and correspondingly for other boundaries) The University of Texas at Austin 7

18 Department of Computer Sciences Graphics Fall 25 (Lecture ) Rendering via Subdivision: Divide up into polygons:. By stepping s =, δ, 2δ,..., t =, γ, 2γ,..., and joining up sides and diagonals to produce a triangular mesh 2. By subdividing and rendering the control polygon The University of Texas at Austin 8

19 Department of Computer Sciences Graphics Fall 25 (Lecture ) Subdivision for olyhedra Regular olyhedra (latonic Solids) Tetrahedron Octahedron Icosahedron Hexahedron (Cube) Dodecahedron The University of Texas at Austin 9

20 Department of Computer Sciences Graphics Fall 25 (Lecture ) Catmull Clark Refinement rule used by Catmull-Clark subdivision scheme is as follows. New vertices are added on each edge and in the center. When connected, 4 new level j + quadrilaterals are produced from the single level j quadrilateral. V j V j 8 V j 7 E j+ 2 V j 2 V j V j 6 E j+ 3 F j+ E j+ E j+ V j+ V j 3 V j 4 V j 5 Catmull-Clark subdivision scheme. Circles are the j level and Squares are the j + level. The vertex rule, edge rule and face rule are shown in the following figure. Each black circle The University of Texas at Austin 2

21 Department of Computer Sciences Graphics Fall 25 (Lecture ) represents a vertex at level j; we compute the position of the vertex at level j + marked by the black square. Note that for the vertex rule, the control vertex with weight 9 6 and the new vertex aren t necessarily aligned as they are in the figure. V j 64 V j V j 7 V j 6 6 V j 8 V j 4 4 V j 8 V j V j V j 6 V j E j+ 3 8 V j F j V j 3 V j 4 V j 5 V j 3 V j 4 V j 2 V j Vertex rule: V j+ = 9 6 V j (V j 2 + V j 4 + V j 6 + V j 8 ) + 64 (V j + V j 3 + V j 5 + V j 7 ) The University of Texas at Austin 2

22 Department of Computer Sciences Graphics Fall 25 (Lecture ) Edge rule: Face rule: E j+ = 3 8 (V j + V j 2 ) + 6 (V j + V j 3 + V j 4 + V j 8 ) F j+ = 4 (V j + V j 2 + V j + V j 8 ) Arbitrary Meshes We have defined Catmull-Clark scheme on quadrilaterals; it can be extended to handle arbitrary polygonal meshes. Observe that if we do one step of refinement, splitting each edge into two and inserting a new vertex for each face (see below Figure), we get a mesh which has only quadrilateral faces. On all other steps of subdivision standard rule described above can be applied. Splitting a hexagon into quadrilaterals. The University of Texas at Austin 22

23 Department of Computer Sciences Graphics Fall 25 (Lecture ) Reading Assignment and News Chapter pages , of Recommended Text. lease also track the News section of the Course Web ages for the most recent Announcements related to this course. ( The University of Texas at Austin 23