PROBLEMS
20.1. Draw the orthographic and spherical perspective aspect graphs of the transparent Flatland object below along with the corresponding aspects.
Solution The visual events for the transparent object, along with the various cells of the perspective and orthographic aspect graph, are shown below.
7
6
2 1 12
11 9
8
3 10
5 1 4
2 3 4
5 9 7
10
13
14 15
16 17
8
12 11
18 19
20
6
Note that cells of the perspective aspect graph created by the intersection of visual event rays outside of the box are not shown. Three of the perspective aspects are shown below. Note the change in the order of the contour points between aspects 1 and 13, and the addition of two contour points as one goes from aspect 1 to aspect 12.
20.2. Draw the orthographic and spherical perspective aspect graphs of the opaque object along with the corresponding aspects.
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Solution The visual events for the opaque object, along with the various cells of the perspective and orthographic aspect graph, are shown below.
7
3 2 1
8 6
4 5
1
2
4
3 8
5
7 6
9
10 11
13 12
14
Note that cells of the perspective aspect graph created by the intersection of visual event rays outside of the box are not shown. Three of the perspective aspects are shown below. Note the change in the order of the contour points between aspects 1 and 13, and the addition of a single contour point as one goes from aspect 1 to aspect 12.
20.3. Is it possible for an object with a single parabolic curve (such as a banana) to have no cusp of Gauss at all? Why (or why not)?
Solution No it is not possible. To see why, consider a nonconvex compact solid.
It is easy to see that the surface bounding this solid must have at least one convex point, so there must exist a parabolic curve separating this point from the noncon-vex part of the surface. On the Gaussian sphere, as one crosses the image of this curve (the fold) at some point, the multiplicity of the sphere covering goes from k to k + 2 with k≥ 1, or from k to k − 2 with k ≥ 3. Let us choose the direction of traversal so we go from k to k + 2. Locally, the fold separates layers k + 1 and k + 2 of the sphere covering (see diagram below).
Layer k+2
Layer
k k+2 k
k
Layer k+1
If there is no cusp, the fold is a smooth closed curve that forms (globally) the boundary between layers k + 1 and k + 2 (the change in multiplicity cannot change along the curve). But layer k + 1 must be connected to layer k by another fold curve. Thus either the surface of a nonconvex compact solid admits cusps of Gauss, or it has at least two distinct parabolic curves.
88 Chapter 20 Aspect Graphs
20.4. Use an equation-counting argument to justify the fact that contact of order six or greater between lines and surfaces does not occur for generic surfaces. (Hint:
Count the parameters that define contact.)
Solution A line has contact of order n with a surface when all derivatives of order less than or equal to n− 1 of the surface are zero in the direction of the line. Ordinary tangents have order-two contact with the surface, and there is a three-parameter family of those (all tangent lines in the tangent planes of all surface points); asymptotic tangents have order-three contact and there is a two-parameter family of those (the two asymptotic tangents at each saddle-shaped point); order-four contact occurs for the asymptotic tangents along flecnodal and parabolic curves; there are a finite number of order-five tangents at isolated points of the surface (including gutterpoints and cusps of Gauss); and finally there is in general no order-six tangent.
20.5. We saw that the asymptotic curve and its spherical image have perpendicular tan-gents. Lines of curvature are the integral curves of the field of principal directions.
Show that these curves and their Gaussian image have parallel tangents.
Solution Let us consider a line of curvature Γ parameterized by its arc length.
Its unit tangent at some point P is by definition a principal direction eiin P . Let κi denote the corresponding principal curvature. Since principal directions are the eigenvectors of the differential of the Gauss map, the derivative of the unit surface normal along Γ is dN ei= κiei. This is also the tangent to the Gaussian image of the principal curve, and the result follows.
20.6. Use the fact that the Gaussian image of a parabolic curve is the envelope of the asymptotic curves intersecting it to give an alternate proof that a pair of cusps is created (or destroyed) in a lip or beak-to-beak event.
Solution Lip and beak-to-beak events occur when the Gaussian image of the oc-cluding contour becomes tangent to the fold associated with a parabolic point. Let us assume that the fold is convex at this point (a similar reasoning applies when the fold is concave, but the situation becomes more complicated at inflections). There exists some neighborhood of the tangency point such that any great circle inter-secting the fold in this neighborhood will intersect it exactly twice. As illustrated by the diagram below, two of the asymptotic curve branches tangent to the fold at the intersections admit a great circle bitangent to them.
circles Great Gauss map Fold of the Gaussian image of asymptotic curves
This great circle also intersects the fold exactly twice, and since it is tangent to the asymptotic curves, it is orthogonal to the corresponding asymptotic direction. In other words, the viewing direction is an asymptotic direction at the corresponding points of the occluding contour, yielding two cusps of the image contour.
20.7. Lip and beak-to-beak events of implicit surfaces. It can be shown (Pae and Ponce, 2001) that the parabolic curves of a surface defined implicitly as the zero set of some density function F (x, y, z) = 0 are characterized by this equation and P (x, y, z) = 0,
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It can also be shown that the asymptotic direction at a parabolic point isA∇F . (a) Show thatAH = Det(H)Id, where H denotes the Hessian of F .
(b) Show that cusps of Gauss are parabolic points that satisfy the equation∇PTA∇F = 0. Hint: Use the fact that the asymptotic direction at a cusp of Gauss is tan-gent to the parabolic curve, and that the vector∇F is normal to the tangent plane of the surface defined by F = 0.
(c) Sketch an algorithm for tracing the lip and beak-to-beak events of an implicit surface.
Solution
(a) Note that the Hessian can be written as
H =¡
But, if ai denotes the ith coordinate of a, we can rewriteA as
A =
since the determinant of the matrix formed by three vectors is the dot product of the first vector with the cross product of the other two vectors.
(b) The parabolic curve can be thought of as the intersection of the two surfaces defined by F (x, y, z) = 0 and P (x, y, z) = 0. Its tangent lies in the intersection of the tangent planes of these two surfaces and is therefore orthogonal to the normals ∇F and ∇P . For a point to be a cusp of Gauss, this tangent must be along the asymptotic direction A∇F , and we must therefore have
∇FTA∇F = 0, which is automatically satisfied at a parabolic point, and
∇PTA∇F = 0, which is the desired condition.
(c) To trace the lip and beak-to-beak events, simply use Algorithm 20.2 to trace the parabolic curve defined inR3by the equations F (x, y, z) = 0 and P (x, y, z) = 0, computing for each point along this curve the vectorA∇F as the corre-sponding asymptotic direction. The cusps of Gauss can be found by adding
∇PTA∇F = 0 to these two equations and solving the corresponding system of three polynomial equations in three unknowns using homotopy continua-tion.
90 Chapter 20 Aspect Graphs
20.8. Swallowtail events of implicit surfaces. It can be shown that the asymptotic direc-tions a at a hyperbolic point satisfy the two equadirec-tions∇F · a = 0 and aTHa = 0, where H denotes the Hessian of F . These two equations simply indicate that the order of contact between a surface and its asymptotic tangents is at least equal to three. Asymptotic tangents along flecnodal curves have order-four contact with the surface, and this is characterized by a third equation, namely
aTHxa aTHya aTHza
· a = 0.
Sketch an algorithm for tracing the swallowtail events of an implicit surface.
Solution To trace the swallowtail events, one can use Algorithm 20.2 to trace the curve defined in R6 by F (x, y, z) = 0, the three equations given above, and
|a|2 = 1. There are of course six unknowns in that case, namely x, y, z, and the three coordinates of a. The corresponding visual events are then given by the values of a along the curve. Alternatively, one can use a computer algebra system to eliminate a among the three equations involving it. This yields an equation S(x, y, z) = 0, and the flecnodal curve can be traced as the zero set of F (x, y, z) and S(x, y, z). The corresponding visual events are then found by computing, for each flecnodal point, the singular asymptotic directions as the solutions of the three equations given above. Since these equations are homogeneous, a can be computed from any two of them. The singular asymptotic direction is the common solution of two of the equation pairs.
20.9. Derive the equations characterizing the multilocal events of implicit surfaces. You can use the fact that, as mentioned in the previous exercise, the asymptotic direc-tions a at a hyperbolic point satisfy the two equadirec-tions∇F · a = 0 and aTHa = 0.
Solution Triple points are characterized by the following equations in the posi-tions of the contact points xi= (xi, yi, zi)T (i = 1, 2, 3):
( F (x
i) = 0, i = 1, 2, 3,
(x1− x2)· ∇F (xi) = 0, i = 1, 2, 3, (x2− x1)× (x3− x1) = 0.
Note that the vector equation involving the cross product is equivalent to two independent scalar equations, thus triple points correspond to curves defined inR9 by eight equations in nine unknowns.
Tangent crossings correspond to curves defined inR6by the following five equations in the positions of the contact points x1and x2:
( F (x
i) = 0, i = 1, 2,
(x1− x2)· ∇F (xi), i = 1, 2, (∇F (x1)× ∇F (x2))· (x2− x1) = 0.
Finally, cusps crossings correspond to curves defined in R6 by the following five equations in the positions of the contact points x1 and x2:
F (xi) = 0, i = 1, 2,
(x1− x2)· ∇F (xi) = 0, i = 1, 2, (x2− x1)TH(x1)(x2− x1) = 0,
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where the last equation simply expresses the fact that the viewing direction is an asymptotic direction of the surface in x1.
Programming Assignments
20.10. Write a program to explore multilocal visual events: Consider two spheres with different radii and assume orthographic projection. The program should allow you to change viewpoint interactively as well as explore the tangent crossings associated with the limiting bitangent developable.
20.11. Write a similar program to explore cusp points and their projections. You have to trace a plane curve.