TNM079 Modeling & Animation Lecture 6 (Implicit surfaces)

Transcription

1 TNM079 Modeling & Animation Lecture 6 (Implicit surfaces) Mark Eric Dieckmann, Media and Information Technology, ITN Linköpings universitet Campus Norrköping SE Norrköping May 4, 2016

2 Content of the sixth lecture We have previously discussed explicit surface definitions: Polygonal meshes: represented by polygons. Subdivision surfaces: represented by polynomial patches. Here we discuss implicit surfaces and topics like: Boolean operations for implicit surfaces. Density functions. Quadric surfaces in local and world coordinate systems. Ray intersections for quadrics.

3 Implicit surfaces Consider a scalar field that is defined in a 3D cartesian space F (x, y, z). Every position vector x = (x, y, z) is connected to a scalar value, e.g. temperature, density etc. ( ) Left: F (x, y, z) = 1 8 (x 0.5) 2 + 2(y 0.5) 2 + (z 0.5) 2 /2 Implicit surfaces are defined by a constraint F ( x) = c. Right: c = 0.

4 Aspects of implicit surfaces In what follows we consider the zero-contour F ( x) = 0: The zero-contour separates the exterior with F ( x) > 0 from the interior with F ( x) < 0 (standard convention). Implicit surfaces are closed: they have no holes. Implicit surfaces are manifold surfaces (no (self-)intersections). Implicit surfaces can be defined straightforwardly only for relatively simple shapes (primitives). Implicit surfaces can be combined by boolean operations. Boolean operations are union, intersection and difference.

5 Normal and signed distance function The gradient F ( x) of the scalar function F ( x) at x F ( x) = e x x F ( x) + e y y F ( x) + e z z F ( x). points along the normal and along increasing values of F ( x). The surface normal is n = F ( x)/ F ( x). Eikonal equation: The gradient of the function F ( x) = 1 everywhere. F ( x) gives us in this case the distance of the point x from the implicit surface F ( x) = 0. The Eikonal equation will turn out to be important in particular when we start moving the zero-countour F ( x) = 0.

6 Quadric surfaces We have previously discussed polynomial (spline) surfaces. For now we don t need deformable surfaces and we don t use splines. Quadric surfaces are polynomials that are used for implicit surfaces. As the name says, they are built upon quadratic polynomials in x, y, z. These polynomials contain terms x, x 2, y, y 2 and z, z 2 and involve also mixed terms and one constant: F (x, y, z) = Ax 2 +2Bxy +2Cxz +2Dx +Ey 2 +2Fyz +2Gy +Hz 2 +2Iz +J. The implicit surface is the zero-contour F (x, y, z) = 0.

7 Examples of quadric surfaces Many quadric surfaces F ( x) = 0 exist with: F (x, y, z) = Ax 2 +2Bxy +2Cxz +2Dx +Ey 2 +2Fyz +2Gy +Hz 2 +2Iz +J A complete overview is given on: We discuss a few of them. We can model an infinite plane by setting A = B = C = E = F = H = 0. F (x, y, z) = 2Dx + 2Gy + 2Iz + J = 0 or (D, G, I )(x, y, z) t + J/2 = 0. This is the equation of a plane with the normal (D, G, I ) and where a point on the plane fulfills (D, G, I )(x 0, y 0, z 0 ) t = J/2.

8 Examples of quadric surfaces Infinite cylinder: B = C = D = F = G = H = I = 0. F (x, y, z) = Ax 2 + Ey 2 + J = 0 or Ax 2 + Ey 2 = J. You recover the equation for an axis-symmetric cylinder by setting A = E and by taking a radius r with J = Ar 2 : x 2 + y 2 = r 2. Ellipsoid: B = C = D = F = G = I = 0. F (x, y, z) = Ax 2 + Ey 2 + Hz 2 + J = 0. Special case: A = E = H and J = Ar 2 : Sphere: x 2 + y 2 + z 2 = r 2. Double cone: B = C = D = F = G = I = J = 0 F (x, y, z) = Ax 2 + Ey 2 Hz 2 = 0. Example: A = E = H: x 2 + y 2 = z 2 resembles the equation of the infinite cylinder but the radius is r = z.

9 Quadratic form The quadric surface is F (x, y, z) = Ax 2 + Bxy + Cxz + Dx + Bxy + Ey 2 + Fyz + Gy + Cxz + Fyz + Hz 2 + Iz + Dx + Gy + Iz + J which we transform with the vertex x = (x, y, z, 1) into the quadratic form A B C D x F (x, y, z) = (x, y, z, 1) B E F G y C F H I z D G I J 1

10 Normals of quadric surfaces The surface of a quadric can be expressed with F (x, y, z) = 0 as: A B C D x F (x, y, z) = (x, y, z, 1) B E F G y C F H I z D G I J 1 This is a quadratic form F (x, y, z) = xq x t. Ideally we would also like to express the gradient (normal) n = such a form. The gradient of the implicit surface F (x, y, z) is computed as F (x, y, z) = ( x F (x, y, z), y F (x, y, z), z F (x, y, z)). F F in

11 Normals of quadric surfaces The normal direction of an implicit surface is given by F (x, y, z) = ( x F (x, y, z), y F (x, y, z), z F (x, y, z)). The implicit surface is defined by F (x, y, z) = Ax 2 +2Bxy +2Cxz +2Dx +Ey 2 +2Fyz +2Gy +Hz 2 +2Iz +J. Ax + By + Cz + D F (x, y, z) = 2 Bx + Ey + Fz + G = Cx + Fy + Hz + I x A B C D = 2 B E F G y z C F H I 1

12 Affine transforms of quadric surfaces The implicit surface can be expressed in the quadratic form F (x, y, z) = xq x t = 0 in the local system. The quadric is defined in its local coordinate system. We can transform a vertex x into the world system through x t = M x t. The world coordinate is then related to the local one via x t = M 1 x t and the quadratic form becomes: F (x, y, z) = ( M 1 x t) t Q ( M 1 x t) = x ˆQ x t. The transformed matrix is ˆQ = M 1t Q M 1.

13 Ray intersections for quadrics The implicit surface is defined in in the world system is x ˆQ x t. Let a ray start at the vertex x 0 and go into the direction r. Its equation is x (t) = x 0 + t r with t 0. We convert x (t) into homogeneous coordinates and substitute it into the function x ˆQ x t. We get a quadratic equation for t. Three possibilities exist: No intersection: complex solutions for t. One (two) intersection(s): Two equal (different) real solutions for t.

14 Boolean operations One implicit surface F ( x) = 0 is not enough to model complicated objects. We can merge multiple surfaces or obtain new ones with the help of the Boolean operators union, intersection and difference. These operators combine the functions and the surface is defined by this new function. The symbols for these operations are: Union, Intersection and difference. The expression max(f 1, F 2 ) selects at each point x the larger value of F 1 ( x) or F 2 (x). The expression min(f 1, F 2 ) selects at each point x the smaller value of F 1 ( x) or F 2 (x).

15 Union operator We apply the union operator to the two objects A and B with perimeters defined by F A ( x) = 0 and F B ( x) = 0. F < 0 A F < 0 B A B The new function is defined by F A B = min(f A, F B ). The perimeter enwrapping F A B = 0 is drawn in black.

16 Intersection operator We apply the intersection operator to the two objects A and B with perimeters defined by F A ( x) = 0 and F B ( x) = 0. F < 0 A F < 0 B A B The new function is defined by F A B = max(f A, F B ). The perimeter F A B = 0 is drawn in black.

17 Difference operator We apply the difference operator to the two objects A and B with perimeters defined by F A ( x) = 0 and F B ( x) = 0. F < 0 A F < 0 B A B The new function is defined by F A B = max(f A, F B ). The perimeter F A B = 0 is drawn in black.

18 Problems with Boolean operations Consider the objects A and B (left) and the one from their union. The perimeters are defined by F A ( x) = 0, F B ( x) = 0 and F A B = 0. A A B B The perimeters are reproduced correctly, but F A B is no longer a signed distance function (gradient condition not fulfilled).

19 Problems with Boolean operations Typically implicit surfaces are smooth, for example because they are higher order polynomials. Shading A and B separately would give smooth shades. A B Cusp The surface defined by the union A B has a cusp. Typically the surface is only C 0 (continuous) along the intersection curve.

20 Density functions ( fuzzy objects) Pick a smooth density function D(F ( x)) with: D(F ( x)) > 1 if x is inside the surface. D(F ( x)) = 1 if x is on the surface. D(F ( x)) ɛ [0, 1[ if x is outside the surface. Negative inside convention A = { x ɛ R 3 F A ( x) 0 }. The signed distance function is mapped to a density function as D A (F A ( x)) = exp ( F A ( x)). We define an exact Boolean according to: D A B = max(d A, D B ) = lim p (D p A + Dp B )1/p. D A B = min(d A, D B ) = lim p (D p A + Dp B )1/p. At small p an approximate Boolean is obtained.

21 Super-elliptic blends The approximate Booleans permit us to obtain smooth connections between implicit surfaces. 0<D(x)<1 D(x) = 1 D(x) > 1 The contour D A B ( x) = (D p A + Dp B )1/p = 1 will smoothly connect both objects (blue curve). Problem: The visual appearance of the merged objects can not be controlled intuitively.

22 Summary Implicit surfaces have been presented as an alternative to polygonal meshes and subdivision surfaces. We have discussed the signed distance function as a means to define a surface. Quadric surfaces have been discussed, as well as how we can render and transform them. Complicated implicit surfaces can be generated by combining elementary ones with the help of Boolean operations. Problems appear related to the signed distance function and the surface smoothness. Smoothness can be achieved through density functions, e.g. super-elliptic blends.