CMSC427 Parametric surfaces (and alternatives)
Generating surfaces • From equations • From data • From curves • Extrusion • Straight • Along path • Lathing (rotation) • Lofting 2
Constructive Solid Geometry (CSG) • Alternative/supplement to parametric shapes • Vocabulary: • Basic set of shapes (sphere, box, cylinder, etc) • Set operations on shapes • Union • Intersection • Difference • Demo • Tinkercad 3
Constructive Solid Geometry (CSG) • Computer Aided Design (CAD) • Precise 3D modeling for industrial design • Less freeform, more control and feedback on shapes • Often compiled (openScad.org) cube([2,3,4]); translate([3,0,0]) { cube([2,3,4]); } color([1,0,0]) cube([2,3,4]); translate([3,0,0]) color([0,1,0]) cube([2,3,4]; translate([6,0,0]) color([0,0,1]) cube([2,3,4]); 4
POVRAY • Stale but interesting ray tracing software • Scene description language (SDL) • Pixar’s Renderman #include "colors.inc" background { color Cyan } camera { location <0, 2, -3> look_at <0, 1, 2> } sphere { <0, 1, 2>, 2 texture { pigment { color Yellow } } } light_source { <2, 4, -3> color White } 5
POVRAY • Support CSG operations union { box { <1, 1, 1>, <2, 2, 2> } sphere{ <1.5, 1.5, 1.5>, 1 } } 6
Piecewise Bézier curves • Each segment spans four control points • Each segment contains four Bernstein polynomials • Each control point belongs to one Bernstein polynomial x 2 ( t ) x 0 ( t ) x 1 ( t ) x 3 ( t ) u = 12 u = 0 Bernstein polynomials u Segment 0 3 6 9 12 7
Curved surfaces Curves • Described by a 1D series of control points • A function x ( t ) • Segments joined together to form a longer curve Surfaces • Described by a 2D mesh of control points • Parameters have two dimensions (two dimensional parameter domain) • A function x ( u,v ) • Patches joined together to form a bigger surface 8
Parametric surface patch • x ( u,v ) describes a point in space for any given ( u,v ) pair • u,v each range from 0 to 1 x ( 0.8,0.7 ) v z 1 u y v x 0 1 u 2D parameter domain 9
Parametric surface patch • x ( u,v ) describes a point in space for any given ( u,v ) pair • u,v each range from 0 to 1 x (0.8,0.7) v x (0.4 ,v ) x ( u, 0.25) z 1 u y v x 0 1 u • Parametric curves 2D parameter domain • For fixed u 0 , have a v curve x ( u 0 ,v ) • For fixed v 0 , have a u curve x ( u,v 0 ) • For any point on the surface, there is one pair of parametric curves that go through point 10
Tangents • The tangent to a parametric curve is also tangent to the surface • For any point on the surface, there are a pair of (parametric) tangent vectors • Note: not necessarily perpendicular to each other ¶ x ¶ v v ¶ x ¶ u u 11
Tangents Notation • Tangent along u direction or or • Tangent along v direction or or • Tangents are vector valued functions, i.e., vectors! 12
Surface normal • Cross product of the two tangent vectors • Order matters (determines normal orientation) • Usually, want unit normal • Need to normalize by dividing through length ¶ x !" n ¶ v ¶ x ¶ u 13
Bilinear patch • Control mesh with four points p 0 , p 1 , p 2 , p 3 • Compute x ( u,v ) using a two-step construction p 2 p 3 v p 0 p 1 u 14
Bilinear patch (step 1) • For a given value of u , evaluate the linear curves on the two u - direction edges • Use the same value u for both: q 1 = Lerp(u, p 2 , p 3 ) q 1 p 2 p 3 v p 0 q 0 p 1 u q 0 = Lerp ( u, p 0 , p 1 ) 15
Bilinear patch (step 2) • Consider that q 0 , q 1 define a line segment • Evaluate it using v to get x x = Lerp ( v , q 0 , q 1 ) q 1 p 2 p 3 x v p 0 q 0 p 1 u 16
Bilinear patch • Combining the steps, we get the full formula x ( u , v ) = Lerp ( v , Lerp ( u , p 0 , p 1 ), Lerp ( u , p 2 , p 3 )) q 1 p 2 p 3 x v p 0 q 0 p 1 u 17
Bilinear patch • Try the other order • Evaluate first in the v direction r 0 = Lerp ( v , p 0 , p 2 ) r 1 = Lerp ( v , p 1 , p 3 ) p 2 p 3 r 0 r 1 v p 0 p 1 u 18
Bilinear patch • Consider that r 0 , r 1 define a line segment • Evaluate it using u to get x x = Lerp ( u , r 0 , r 1 ) p 2 p 3 x r 0 r 1 v p 0 p 1 u 19
Bilinear patch • The full formula for the v direction first: x ( u , v ) = Lerp ( u , Lerp ( v , p 0 , p 2 ), Lerp ( v , p 1 , p 3 )) p 2 p 3 x r 0 r 1 v p 0 p 1 u 20
Bilinear patch • It works out the same either way! x ( u , v ) = Lerp ( v , Lerp ( u , p 0 , p 1 ), Lerp ( u , p 2 , p 3 )) x ( u , v ) = Lerp ( u , Lerp ( v , p 0 , p 2 ), Lerp ( v , p 1 , p 3 )) q 1 p 2 p 3 x r 0 r 1 v p 0 q 0 p 1 u 21
Bilinear patch • Visualization 22
Bilinear patches • Weighted sum of control points • Bilinear polynomial • Matrix form exists, too 23
Properties • Interpolates the control points • The boundaries are straight line segments • If all 4 points of the control mesh are co-planar, the patch is flat • If the points are not coplanar, get a curved surface • saddle shape, AKA hyperbolic paraboloid • The parametric curves are all straight line segments! • a (doubly) ruled surface: has (two) straight lines through every point • Not terribly useful as a modeling primitive 24
Bicubic B é zier patch • Grid of 4x4 control points, p 0 through p 15 • Four rows of control points define B é zier curves along u p 0 ,p 1 ,p 2 ,p 3 ; p 4 ,p 5 ,p 6 ,p 7 ; p 8 ,p 9 ,p 10 ,p 11 ; p 12 ,p 13 ,p 14 ,p 15 • Four columns define B é zier curves along v p 0 ,p 4 ,p 8 ,p 12 ; p 1 ,p 6 ,p 9 ,p 13 ; p 2 ,p 6 ,p 10 ,p 14 ; p 3 ,p 7 ,p 11 ,p 15 p 12 p 13 p 8 v p 9 p 14 p 4 p 15 p 5 p 0 p 10 p 1 p 11 p 6 u p 7 p 2 p 3 25
Bicubic B é zier patch (step 1) • Evaluate four u -direction B é zier curves at u • Get intermediate points q 0 … q 3 q 0 = Bez ( u , p 0 , p 1 , p 2 , p 3 ) q 1 = Bez ( u , p 4 , p 5 , p 6 , p 7 ) q 2 = Bez ( u , p 8 , p 9 , p 10 , p 11 ) q 3 = Bez ( u , p 12 , p 13 , p 14 , p 15 ) p 12 q 3 p 13 p 8 v p 9 p 14 p 4 p 15 p 5 q 2 q 1 p 0 p 10 p 1 p 11 p 6 q 0 u p 7 p 2 p 3 26
Bicubic B é zier patch (step 2) • Points q 0 … q 3 define a B é zier curve • Evaluate it at v x ( u , v ) = Bez ( v , q 0 , q 1 , q 2 , q 3 ) p 12 q 3 p 13 p 8 v x p 9 p 14 p 4 p 15 p 5 q 2 q 1 p 0 p 10 p 1 p 11 p 6 q 0 u p 7 p 2 p 3 27
Bicubic B é zier patch • Same result in either order (evaluate u before v or vice versa) q 0 = Bez ( u , p 0 , p 1 , p 2 , p 3 ) r 0 = Bez ( v , p 0 , p 4 , p 8 , p 12 ) q 1 = Bez ( u , p 4 , p 5 , p 6 , p 7 ) 1 = Bez ( v , p 1 , p 5 , p 9 , p 13 ) r q 2 = Bez ( u , p 8 , p 9 , p 10 , p 11 ) r 2 = Bez ( v , p 2 , p 6 , p 10 , p 14 ) Û q 3 = Bez ( u , p 12 , p 13 , p 14 , p 15 ) r 3 = Bez ( v , p 3 , p 7 , p 11 , p 15 ) x ( u , v ) = Bez ( v , q 0 , q 1 , q 2 , q 3 ) x ( u , v ) = Bez ( u , r 0 , r 1 , r 2 , r 3 ) p 12 q 3 p 13 r 0 r 1 p 8 v x p 9 p 14 p 4 p 15 p 5 q 2 r 2 r 3 q 1 p 0 p 10 p 1 p 11 p 6 q 0 u p 7 p 2 p 3 28
T ensor product formulation • Corresponds to weighted average formulation • Construct two-dimensional weighting function as product of two one-dimensional functions • Bernstein polynomials B i , B j as for curves • Same tensor product construction applies to higher order Bézier and NURBS surfaces 29
Bicubic Bézier patch: properties • Convex hull: any point on the surface will fall within the convex hull of the control points • Interpolates 4 corner points • Approximates other 12 points, which act as “handles” • The boundaries of the patch are the Bézier curves defined by the points on the mesh edges • The parametric curves are all B é zier curves 30
Tangents of B é zier patch • Remember parametric curves x ( u,v 0 ), x ( u 0 ,v ) where v 0 , u 0 is fixed • Tangents to surface = tangents to parametric curves • Tangents are partial derivatives of x ( u,v ) • Normal is cross product of the tangents ¶ x p 13 p 12 ¶ v v 0 p 8 v x p 14 p 9 p 4 ¶ x p 15 p 5 ¶ u p 0 p 10 p 1 p 11 p 6 u 0 u p 7 p 2 31 p 3
Tangents of B é zier patch q 0 = Bez ( u , p 0 , p 1 , p 2 , p 3 ) r 0 = Bez ( v , p 0 , p 4 , p 8 , p 12 ) q 1 = Bez ( u , p 4 , p 5 , p 6 , p 7 ) 1 = Bez ( v , p 1 , p 5 , p 9 , p 13 ) r q 2 = Bez ( u , p 8 , p 9 , p 10 , p 11 ) r 2 = Bez ( v , p 2 , p 6 , p 10 , p 14 ) q 3 = Bez ( u , p 12 , p 13 , p 14 , p 15 ) r 3 = Bez ( v , p 3 , p 7 , p 11 , p 15 ) ¶ x ¶ x ¶ v ( u , v ) = Be ¢ ¶ u ( u , v ) = Be ¢ z ( v , q 0 , q 1 , q 2 , q 3 ) z ( u , r 0 , r 1 , r 2 , r 3 ) ¶ x p 13 ¶ v p 12 r 0 q 3 r 1 p 8 v x p 14 p 9 p 4 ¶ x p 15 p 5 q 2 ¶ u r 2 r 3 q 1 p 0 p 10 p 1 p 11 p 6 q 0 u p 7 p 2 32 p 3
T essellating a B é zier patch • Uniform tessellation is most straightforward • Evaluate points on uniform grid of u , v coordinates • Compute tangents at each point, take cross product to get per- vertex normal • Draw triangle strips (several choices of direction) • Adaptive tessellation/recursive subdivision • Potential for “cracks” if patches on opposite sides of an edge divide differently • Tricky to get right, but can be done 33
Recommend
More recommend
