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700/010 CIS Lecture IX Computational Geometry: Distance Fields Suresh Venkatasubramanian Scribed by IkkJin Ahn 1 De nition Given a set of objects, a distance eld is de ned at each point by the smallest distance from the point to the objects. Each object may be represented as data on a voxel grid or as an explicit surface representation. Moreover, the distances between the point and an object can be speci ed using di erent metrics, including Euclidean or max-norm distance. D(p) = min d(p, s) s S = min s S (x1 x1 )2 + (y1 y2 )2 (p = (x1 , y1 ), s = (x2 , y2 )) 2 Application Distance elds are frequently used in computer graphics, geometric modeling, robotics and scienti c visualization. 2.1 Robotics One way of motion planning is calculating all distances from obstacles and maximize the minimum distance from obstacles for each point, i.e., maximin. Figure 1: Planning in an assembly environment: Constraint based planning in a dynamic environment consisting of 26.9k polygons using distance elds. 1 2.2 Computer graphics Figure 2: Cassini Model: A volume rendering of the distance eld of the Cassini with 93K polygons. The distance to the surface is color coded, increasing from red to green to blue. 2.3 Particle systems Figure 3: Motion planning of falling particles. Sites are avoided by using the Voronoi diagram s distance bu er to create a potential eld. This same principle is used in the rigid-body planner. 3 3.1 GPU implementation Basic idea We can get ideas from Vornoi Diagram and Dirichlet domains. The main idea of our approach is to render a polygonal mesh approximation to each site s distance function. Each site is assigned a unique color ID, 2 and the corresponding distance mesh is rendered in that color using a parallel projection. We make use of two components of the graphics hardware: linear interpolation across polygons and the Z-bu er depth comparison operation. When rendering a polygonal distance mesh, the polygon rasterization reconstructs all distances across the mesh. The Zbu er depth test compares the new depth value to the previously stored value. If the new value is less, the Z-bu er records the new distance, and the color bu er records the site s ID. In this way, each pixel in the frame bu er will have a color corresponding to the site to which it is closest, and the depth-bu er will have the distance to that site. 3.2 2D case (Points) The distance function for a point in the plane is a right circular cone.(i.e.,45 cones along Z-direction) We approximate cones as a triangle fan proceeding radially outward from the apex. The fragment of the smallest Z-value represents the value of the distance eld. Question. Why do we use cones? Because Our distance eld is de ned as d(p, s) = PZ , PZ = (px sx )2 + (py sy )2 . (Let PZ as Z coordinate value of the cone). We can use other shape for di erently de ned distance eld. 3.3 2D case (Line segments and polygons) The distance function for a line segment is composed three of parts: one for the segment itself and one for each endpoint. The endpoints are treated the same way as points. The distance function for the line segment is just a tent which is composed of two quadrilaterals. The distance function for polygons can be rendered as a series of linear segments. 3 3.4 3D case (Points) What is di culty in this case? We cannot use 4D cones. Thus, we will introduce sliced spaces which has pz = Z0 . To construct the intersection of the Voronoi diagram with this slice, consider the distance function for a point A, restricted to the slice. Denote the restricted distance function by d(p, s) = dist((px , py , Z0 ), s). D(p) = min d(p, s) s S d(p, s) = d(p, s) = d(p, s) = (px sx )2 + (px sy )2 + (pz sz )2 (px sx )2 + (px sy )2 + (Z0 sz )2 (px sx )2 + (px sy )2 + (Z0 sz )2 If we derive an surface equation from this relation, we can get D(p) = min d(p, s) s S PZ 2 PZ = (px sx )2 + (px sy )2 + c (c = (Z0 sz )2 and PZ is Z coordinate value of certain shape) = (px sx )2 + (px sy )2 + c This shape is hyperboloid of two sheets. The hyperboloid of two sheets looks like two (elliptic) paraboloids facing each other. (If the condition is c < 0, then this shape will be hyperboloid of one sheet.) 4 3.5 3D case (Line segments) Similarly to 2D case, the graph of the distance function for a line segment site is an elliptical cone. The apex of the cone lies at the intersection of the segment s line with the slice, and the cone s eccentricity is determined by the relative angle of the line and the slice. The 3D region of in uence of a line segment lies between two parallel planes through the endpoints, since a point outside these planes is closer to one of the endpoints than to the segment. 5 4 4.1 Errors Distance error - Meshing error OpenGL does not have a function for the exact cone and hyperboloid. Thus, we have to approximate with triangles. Let s suppose the radius of the circle in the bottom of a cone is R, and the angle between the center of the bottom circle and two bottom points of each mesh is . In this case, the relation between the angle and the maximum error is de ned as 2 R R cos = Therefore a maximum distance error of no more than one pixel s width, a cone mesh for a 512x512 grid will require only 60 triangles, and 85 triangles for a 1024x1024 grid. 4.2 Z-bu er errors The limitations of the number of bits of precision provided by the Z-bu er. Current graphics systems have 24 bits or 32 bits of precision for each pixel in the Zbu er, which is more than the 23 bits provided in standard oating-point. If the distances between two pixels cannot be determined within that precision, the Z-bu er cannot accurately choose the correct color. This e ect is small when compared to the other two, but can be signi cant at very high resolutions with very little distance error. A higher-precision Zbu er can be simulated in software at a signi cant loss in e ciency. 6

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