Computational Geometry

Computational GeometryComputational Geometry

- Methods, applications and problems of shape modeling

- Computational geometry Finding Dealunay Triangulations Finding Convex Hulls: Insertion Hull A connection between Delaunay triangulation

and convex hulls Point Enclosure: The Ray-Shooting Method

IntroductionIntroduction

According to Longman Dictionary of Contemporary English the word shape means the outer form of something, that you see or feel. There is no a method or a system that is ideally suited to all tasks and researchers are still focusing on the development of sophisticated algorithms to discover the essential rules of representing and generating shapes.


Computer-aided geometric design (CAGD) mainly based on parametric representation now pervades many areas. In spite of elaborate user interfaces this is very routine work that leads to long training. Constructive solid geometry (CSG) deals with a collection of simple primitives and now is looking the most promising direction in the geometric modeling. However, it suffers from the difficulties connected with free form modeling..

IntroductionIntroduction Polygonal representation is rather popular in animation field. However, we have to notice that polygonal patches, in fact in many cases, are the result of another representations or processing data. Geometric objects defined by real functions of several variables (so-called implicit models), with the help of procedural techniques (physically based modeling, L-systems, particle systems) have proven to be useful in CG, geometric modeling and animation. The main problems of this technique are the performance, interface and interactivity. Physics based modeling essentially is based on the Finite Element Method which has a history in the application of structural analysis of materials. This method is used in CAD applications and different tissue simulations were performed for computer animation goals and surgery planning. This models are computationally expensive.


A model is an artificially constructed object that makes the observation of another object easier. Examples of models are physical models of ships and cars or human faces, molecule models used in chemistry or biology, mathematical models modeled in terms of numerical data and equations. Models are useful because they can present or study characteristics of an object more easily by specifying certain methods or another (underlying) models if they are needed. The key observation is that many problems are inherently geometric and it makes sense to separate the data dealing with the geometric shape from other, non-geometric data called object model.

Computational geometryComputational geometry

Computer graphics encompasses methods for rendering scenes. Modeling is used to construct a scene description. Computational geometry encompasses algorithms for solving geometry problems. The problems we will cover are easy to formulate and invoke simple geometric object: points, lines, polygons. Traditionally, computational geometry includes the definition or statement of a problem, algorithm and an algorithmic paradigm. Algorithms employ objects for organizing data called data structures.

Computational geometry:Computational geometry:Finding Dealunay Finding Dealunay TriangulationsTriangulations

A triangulation of a finite point set S is a triangulation of the convex hull CH(S) that uses all the points of S. The convex hull of set S equals the intersection of all convex polygons which contain S. Equivalently, CH(S) is the convex polygon of minimum area which contains all the points of S. Equivalent definition states that CH(S) equals the union of all triangles determined by points of S. A region in the plane is convex if for any two points in the region, the line segment between the two points lies in the region.

Finding Dealunay TriangulationsFinding Dealunay Triangulations

A polygon is a closed curve in the plane composed of straight line segments. The segments are called the edges or sides of the polygon, and the end points where two segments meet are called its vertex. The number of vertices that a polygon possesses is its size. A polygon is simple if it does not cross itself. A simple polygon enclosed a connected region of the plane, referred to as its interior. The unbounded region surrounding a simple polygon forms its exterior, and the set of points lying on the polygon itself forms its boundary.


Two triangulations of the same set of points.

The edges of a triangulation of S can be classified as hull edges and interior edges. The hull edges are those edges that lie along the boundary of the convex hull CH(S), and the interior edges are the remaining edges, those that pierce the convex hull interior. Observe that every edge of the triangulation is met by two faces: each interior edge by two triangles, and each hull edge by one triangle and the unbounded plane.


Point-Set Triangulation Theorem: Suppose point set S contains n 3 points, not all collinear. Suppose further that i of the points are interior [lying in the interior of CH(S)]. Then every triangulation of S contains exactly n + i – 2 triangles. To see why this theorem is true, first consider triangulating the n – i boundary points. Since they are the vertices of a convex polygon any such triangulation contains (n – i) – 2 triangles. Now consider incorporating the remaining i interior points into the triangulation, one at time. We claim that adding each such point increases the number of triangles by two.


The two cases illustrated in Figure can occur. First, if the point falls in the interior of some triangle, the triangle is replaced by three new triangles.

Second, if the point falls on some edge of the triangulation, each of the two triangles that meet the edge is replaced by two new triangles. It follows that after all i points are inserted, the total number of triangles is (n – i - 2) + (2i), or simply (n + i - 2).


Delaunay triangulations are well balanced in the sense that triangles tend toward equiangularity. Figure shows the Delaunay triangulation of a large point set (250 points chosen at random within a rectangle).


To define Delaunay triangulation, we need some new definitions. A set of points is cocircular if there exists some circle on whose boundary all the points lie. If the circle is unique, it is called the circumcircle of the points. The circumcircle of a triangle is simply the circumcircle of its three (non-collinear) vertices. A circle is said to be point free with respect to a given point set S if none of the points of S lies in the circle`s interior. Points of S may, however, lie along the boundary of a point-free circle. A triangulation of point set S is a Delaunay triangulation if the circumcircle of every triangle is point free.


We will make two assumptions about point set S to simplify the triangulation algorithm. First, to ensure that some triangulation exists, we will assume that S contains at least three points, not all collinear. Second, to ensure the Delaunay triangulation is unique, we will assume that no four points of S are cocircular. Algorithm works by growing a current triangulation, triangle by triangle. In each iteration, the algorithm seeks a new triangle which attaches to the frontier of the current triangulation.


The definition of frontier depends on the following scheme, which classifies the edges of the Delaunay triangulation relative to currentr triangulation. Every edge is either dormant, live, or dead:

1. Dormant edges: An edge of the Delaunay triangulation is dormant if it has not yet been discovered by the algoritm.

2. Live edges: An edge is live if it has been discovered but only one of its faces is known.

3. Dead edges: An edge is dead if it has been discovered and both of its faces are known.


Initially only a single hull edge is live – the unbounded plane is known to meet it – and all remaining edges are dormant. As the algorithm proceeds, edges transition from dormant to live, from live to dead. The frontier at each stage consists of the set of live edges.

In each iteration, we select any edge e of the frontier and process it, which consist of seeking edge e`s unknown face.

If this face turns out to be some triangle t determined by the endpoints of e and some third vertex v, edge e dies since both of its faces are now known. Moreover, each of the other two edges of triangle t transition to the next state: from dormant to live, or from live to dead. Here vertex v is called the mate of edge e.

Alternatively, if the unknown face turn out to be unbounded plane, edge e simply dies. In this case e has no mate.


Figure illustrates the algorithm. In the figure, the action proceeds top to bottom, then left to right. The frontier in each stage is darkened.

A connection between Delaunay triangulation and A connection between Delaunay triangulation and convex hulls.convex hulls.

There is a fascinating relationship between Delaunay triangulations and the convex hull of a particular set of 3D points.

Delaunay triangulations and convex hulls appear to be quite different structures, one is based on metric properties (distances) and the other on affine properties (collinearity, coplanarity).

It is possible to convert the problem of computing a Delaunay triangulation in dimension d to that of computing a convex hull in dimension d + 1.

The connection between the two structures is the paraboloid z = x2 + y2. Observe that this equation defines a surface whose vertical cross sections (constant x or constant y) are parabolas, and whose horizontal cross sections (constant z) are circles. For each point in the plane, (x, y), the vertical projection of this point onto this paraboloid is (x,y, x2 +y2) in 3 space.

A connection between Delaunay triangulation and A connection between Delaunay triangulation and convex hulls. convex hulls. Two-Dimensional Delaunay TriangulationsTwo-Dimensional Delaunay Triangulations

The paraboloid is

z = x2 + y2,

see Figure.

A connection between Delaunay triangulation and A connection between Delaunay triangulation and convex hulls (Cont)convex hulls (Cont)

Given a set of points S in the plane, let S0 denote the projection of every point in S onto this paraboloid.

Consider the lower convex hull of S0. This is the portion of the convex hull of S0 which is visible to a viewer standing at z = -1.

We claim that if we take the lower convex hull of S0, and project it back onto the plane, then we get the Delaunay triangulation of S.

A connection between Delaunay triangulation and A connection between Delaunay triangulation and convex hulls (Cont)convex hulls (Cont)

In particular, let (p, q, r) be elements of S, and let p0 , q0 , r0 denote the projections of these points onto the paraboloid.

Then p0 q0 r0 define a face of the lower convex hull of S0 if and only if pqr is a triangle of the Delaunay triangulation of S. The process is illustrated in the following fFgure.

A connection between Delaunay triangulation and A connection between Delaunay triangulation and convex hulls (Cont.)convex hulls (Cont.)

Take the given sites/points in the plane, and project them upwards until they hit the paraboloid, that is, map every point as follows:

(xi, yi) (xi, yi , x2i + y2

i).

Take the convex hull of this

set of three-dimensional points; see Figure

A connection between Delaunay triangulation and A connection between Delaunay triangulation and convex hulls. convex hulls. Two-Dimensional Delaunay Triangulations (Cont)Two-Dimensional Delaunay Triangulations (Cont) Now discard the “top” faces

of this hull (points having a positive dot product with the z axis vector). Result is a bottom “shell”. Project this to the xy-plaine.

This is the Delaunay triangulation!

See Figure


The question is, why does this work? To see why, we need to establish the connection between the triangles of the Delaunay triangulation and the faces of the convex hull of transform

In particular, recall that Delaunay condition: Three points p, q, r, in S form a

Delaunay triangle if and only if the circumcircle of these points contains no other point of S.

Convex hull condition: Three points p0, q0, r0 in S0 form a face of the convex hull of S0 if and only if the plane passing through p0, q0, and r0 has all the points of S0 lying to one side.

A connection between Delaunay triangulation and A connection between Delaunay triangulation and convex hulls (Cont.)convex hulls (Cont.) The connection we need to establish is between the

emptiness of circumcircles in the plane and the emptiness of halfspaces in 3 space.

Lemma: Consider 4 distinct points p, q, r, s in the plane, and let p0, q0, r0 s0 be their respective projections onto the paraboloid, z = x2 + y2 . The point s lies within the circumcircle of p, q, r if and only if s0 lies on the lower side of the plane passing through p0, q0, r0 .

A connection between Delaunay triangulation and A connection between Delaunay triangulation and convex hulls (Cont.)convex hulls (Cont.) To prove the lemma, first consider an arbitrary (nonvertical)

plane in 3 space, which we assume is tangent to the paraboloid above some point (a, b) in the plane. To determine the equation of this tangent plane, we take derivatives of the equation z = x2 + y2 with respect to x and y, giving:

dz/dx = 2x

dz/dy = 2y

• At the point (a, b, a2 +b2 ) these evaluate to 2a and 2b. It follows that the plane passing through these point has the form

t = 2ax + 2by + k.


• To solve for k we know that the plane passes through

(a, b, a2 + b2 )

so we solve giving

a2+ b2 = 2 a2 + 2 b2 + k.

• Implying that k = - (a2 + b2 ). Thus the plane equation is:

z = 2ax + 2by - (a2 + b2 )


• If we shift the plane upwards by some positive amount r2 we get the plane

z = 2ax + 2by - (a2 + b2 ) + r2 .

• How does this plane intersect the paraboloid? Since the paraboloid is defined by:

z = x2 + y2

we can eliminate z giving

x2 + y2 = 2ax + 2by - (a2 + b2 ) + r2,

which after some simple rearrangements is equal to

(x - a)2 + (y - b)2 = r2 .

• This is just a circle.


• Thus, we have shown that the intersection of a plane with the paraboloid produces a space curve (which turns out to be an ellipse), which when projected back onto the (x, y) coordinate plane is a circle centered at (a, b).


• Thus, we conclude that the intersection of an arbitrary lower halfspace with the paraboloid, when projected onto the (x, y) plane is the interior of a circle.

• Going back to the lemma, when we project the points p, q, r onto the paraboloid, the projected points p0, q0 and r0 define a plane. Since p0, q0 and r0 , lie at the intersection of the plane and paraboloid, the original points p, q, r lie on the projected circle.

• Thus this circle is the (unique) circumcircle passing through these p, q, and r. Thus, the point s lies within this circumcircle, if and only if its projection s0 onto the paraboloid lies within the lower halfspace of the plane passing through .


The shifted plane intersects the paraboloid in a curve (an ellipse) that projects to a circle!

This is illustrated in Figures - Plane for (a, b) =(2, 2) and r = 1

cutting the paraboloid

- The curve of intersection in the xy-plane.

dt4vornew.exe

The Voronoi diagram The Voronoi diagram The Voronoi diagram of a collection of geometric

objects is a partition of space into cells, each of which consists of the points closer to one particular object than to any others.

Finding Convex Hulls: Insertion Hull Finding Convex Hulls: Insertion Hull

Imagine the plane to be a sheet of wood with a nail protruding from every point in S. Now stretch a rubber band around all the nails and then release it, allowing it to snap taut against the nails. The taut rubber band conforms to the convex hull boundary.


A point is a boundary point if it lies in the convex hull boundary, and an interior point if it lies in the convex hull interior.

Those boundary points which form the “corner” vertices of the convex hull are known as extreme points. Equivalently, a boundary point is extreme if it does not lie between any two other points of S.

Incremental insertion approach. Initially, the current hull consists of a single point of S; at completion, when all points have been inserted, the current hull equals CH(S) and we are done.


When a new point s is inserted into the current hull, one of two cases occurs. In the first case, s may lie in the current hull, in which case the current hull does not need to be updated.

In the second case, s lies outside the current hull, requiring that the current hull be modified as in Figure . Through point s can be drawn two supporting lines - tangent to the current hull.

Point Enclosure: The Ray-Shooting Method Point Enclosure: The Ray-Shooting Method

Point enclosure algorithm decides whether a given point a lies inside, outside, or on the boundary of a convex polygon p. In Figure , for example, the interior of the polygon straddle both sides of the edge labeled e.


The main idea of the ray-shooting method is the following: starting from some point far from the polygon, move in a straight line towards a. Along the way we cross the polygon boundary zero or more times: the first time crossing into the polygon, the second time crossing back out, the third time crossing back in once again, and so forth, until arriving at a. In general, every odd-numbered crossing carries us into polygon p, and every even-numbered crossing carries us back out of p.

If we arrive at a having undergone an odd number of crossing, a lies inside p; and if an even number of crossings, a lies outside p.


Transforming this idea into an algorithm turns on two key observations:

First, any ray that originates at the point a to be classified will do. Being free to work with any ray originating at a, we can for simplicity, work with the right horizontal ray originating at a.

The second key observation is that the order of boundary crossing along ray is irrelevant.


Relative to the right horizontal ray , we should distinguish tree types of polygon edges: touching edges, which contain point a; crossing edges, which do not contain point a but which ray crosses; and inessential edges, which ray does not meet at all. For example, in Figure, edge c is crossing edge, edge d is touching edge, and edge e is an inessential edge.

Further reading J. O’Rourke, Computational Geometry in C, Cambridge

University Press, 2nd ed, 1998

Surfaces: Surfaces: Curvature EstimationCurvature Estimation

• In the past 20 years, a number of remarkable algorithms for calculating the curvature of 3D surfaces have been developed.

• Main objective is to classify each point in the surface in a manner that allows the detection of geometric features as well as the initialization of matching algorithms.

• Applications: clothing design and manufacturing, medical research (disorders) and prosthesis design.

Surfaces: Surfaces: Curvature Based Mesh Curvature Based Mesh ImprovementImprovement • It is very important to improve the quality of surface meshes for numerical

simulations, solid mesh generation, and computer graphics applications.• Optimizing the form of the mesh elements it is necessary to preserve new nodes of the mesh as close as possible to a surface approximated by the initial mesh. • In the method discussed here the new location of each node is found using values of principal curvatures in this node. Such procedure allows preserving new mesh very close to the initial surface while improving element quality. • There are two main ways for mesh optimization: modification of the mesh topology by inserting/deleting mesh nodes or edge flipping and node movement methods, commonly called mesh smoothing.

• Good shape of mesh elements is not only the criteria of mesh quality. It is also important to preserve the new nodes as close as possible to the smooth surface

approximated by the initial mesh.

Surfaces: Surfaces: Curvature Based Mesh Curvature Based Mesh ImprovementImprovement • We formulate the problem of mesh improvement in the following way. A

triangular mesh which serves as a representation (discrete approximation) of a smooth surface is given. It is necessary to improve mesh element quality in such

a way that new nodes keep situated on the approximated smooth surface. • Main idea: The new position of each node is found using principal curvatures calculated for each node. • Such procedure allows keeping vertices of the new mesh very close to approximated smooth surface.

Surfaces: Surfaces: Curvature Based Mesh Curvature Based Mesh ImprovementImprovement • Construction of the quadric The Monge form is used. The quadric of the form

is fitted to the nodes in a local coordinate system whose axis is along a

normal at the concerned vertex and whose origin is at that vertex. It is necessary to determine normals at the nodes of the initial mesh. For

that we calculate normals for each triangle and then average them at shared points.

For quadric interpolation least squares method is used.

Surfaces: Surfaces: Curvature Based Mesh Curvature Based Mesh ImprovementImprovement • Construction of the quadric

For each node of the initial mesh: Get the k nearest neighboring nodes. Compute the tangent plane P at concerned node (i.e. the plane

perpendicular to the normal at ). Define an orthonormal coordinate system in P with a node as an

origin and take the normal as a z axis. Find the k coordinates, of the nearest neighboring nodes in the new

coordinate system. Form the system: , where , - required coefficients of the quadric. Solve the system , where is the least square solution, with

classical Gauss method.

Surfaces: Surfaces: Curvature Based Mesh Curvature Based Mesh ImprovementImprovement • Curvature estimation After the quadric

has been found for each node of the mesh the principal curvatures are calculated using notions of classical differential geometry.

The matrix of the first fundamental form for our quadric is written in the form:

The matrix of the second fundamental form is

The eigenvalues of these pair of forms, i.e. of the equation , are the

principal curvatures: maximal and minimal .

Surfaces: Surfaces: Curvature Based Mesh Curvature Based Mesh ImprovementImprovement • Mesh improvement First, “improved” normals which will be as close as possible to the smooth

surface approximated by initial mesh are calculated. To calculate such normals, area-weighted coefficients are used. Consider an oriented triangle mesh. Let us define the “improved” normal in

some node p of the mesh. For each triangle i incident to the concerned node we calculate the weight coefficient .

Here is the area of the triangle that incident to. We calculate normals for each triangle i incident to the concerned node and averaging them at that node with coefficients . Then we normalize 　 these averaged normals.

Surfaces: Surfaces: Curvature Based Mesh Curvature Based Mesh ImprovementImprovement

• Mesh improvement Consider how the algorithm works in 2D case.

In this case the problem is formulated in the following way.

It is necessary to reduce the difference between the lengths of the segments of the given polygonal line keeping the new nodes as close as possible to the curve approximated this polygonal path.

The algorithm is transformed into the following procedure.

For each node of the polygonal line we define the “ordinary” normals similarly to 3D case.

Then if the node is not the end vertex we draw the circumference passing through concerned node and two nodes incident to it.

From the center of the circumference we draw the radius in the direction opposite to the normal defined in.

Obtained point on the circumference is the new position for as shown in Figure.

Surfaces: Surfaces: Curvature Based Mesh Curvature Based Mesh ImprovementImprovement

• Mesh improvement. Experiments

Comparison between images by the algorithm and Laplacian smoothing.

(a) The sphere optimized with CBMI algorithm using “ideal” normals (the difference of the volume from the original model is 0.93%);

(b) The sphere processed with Laplacian smoothing (the difference of the volume from the original model is 1.83%).

Assigment 2Assigment 2Due Lec. 9Due Lec. 9 Develop an Applet or OpenGL program to calculate and draw

the minimum area closing rectangle for a convex polygon• Given a convex polygon P, what is the smallest-box (in terms of surface area)

enclosing P? Technically, given a direction, the extreme points for P can be computed and an enclosing rectangle is therefore constructed.

• The minimum area rectangle enclosing a convex polygon P has a side collinear with an edge of the polygon.

• Illustrating the above result: the four lines of support (red), with one coinciding with an edge, determine the enclosing rectangle (blue).

• A simple algorithm would be to compute for each edge the corresponding rectangle collinear with it and the construction of this rectangle would imply computing the polygon's extreme points for each edge.

Computational Geometry

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geometric modeling

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