Distance fields

The program Multiple perspectives: euclidean, analytic, transformational Build algebraic skills Foreshadow future math concepts Build intuitions (not bad mnemonics…show mine?)

Scalar field Contour map Like a 3d function projected on a plane Slope field Weather map Hiking map (contour map)

Distance field Not my invention Used in image processing (though I don’t know how!) Overview of Signed Distance Fields A signed distance field is represented as a grid sampling of the closest distance to the surface of an object represented as a polygonal model. Usually the convention of using negative values inside the object and positive values outside the object is applied. Signed distance fields are very attractive in computer graphics and related fields. Often they are used for collision detection in cloth animation (Bridson et al. 2003), multi-body dynamics (Guendelman et al. 2003), deformable objects (Fisher and Lin 2001), mesh generation (Molino et al. 2003), motion planning (Hoff et al. 1999), and sculpting (Bærentzen 2001). However, fast and robust signed distance field computation is often either a performance bottleneck, because of high-resolution fields, or a nearly impossible task because of degeneracies in input meshes. Thus, computation can be tedious and time-consuming. For example, a naive implementation on a CPU can take hours, even days, to complete for high-resolution grids (256 3 resolution or greater).

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Distance field Distance to a set of points in plane Soccer goal example What is a distance field for a point? Image What are the contours? What is equation of a contour? Inside/ outside / on contour

insight Distance from a point to a line It’s where a contour is tangent to the line (otherwise it crosses in two places. Draw angle bisector, etc.) Is there a discussion of a similar proof in a common source?

Two points What will the contours look like? What is that other line? What different kinds of contours are there? (3: distinct circles, tangent circles, ‘conjoined circles’) Under what circumstances does each appear? Notice that boundary is not a contour…not all points on boundary are the same distance. What is significance of points on the boundary? Points on this side of boundary closer to this point, etc. I’m asking bc kids have trouble w this / leads into graphing inequalities

Three points Circumcenter Easy to see that where perp bisectors meet is equidistant to all three vertices Divides triangle into three regions…closer to each vertex

More points: Voronoi diagrams Cell phone towers Crystallography more Voronoi game v1 V2 Competition: what strategies are effective

Voronoi continued Writing name When do perp bisectors of four points meet at one point? Other questions Classify quadrilaterals

A line

Two lines

Three lines

Classifying quadrilaterals by angle bisectors

A point and a line