Summer School on Computational Geometry - 2013, IMPA, Rio de Janeiro, RJ, Brazil Computational Geometry: Delaunay Triangulations and Voronoi Diagrams Lecture 3 Prof. Marcelo Ferreira Siqueira DIMAp – UFRN [email protected] Triangulations n Given a finite family, (ai)i I, of points in E , we say that ⇥ (ai)i I is affinely independent if and only if the family of vec- ⇥ tors, (aiaj)j (I i ), is linearly independent for some i I. ⇥ −{ } ⇥ E2 E2 E2 a0 a1 a 0 a0 a1 a2 A. I. A. I. A. I. 2 Triangulations n Given a finite family, (ai)i I, of points in E , we say that ⇥ (ai)i I is affinely independent if and only if the family of vec- ⇥ tors, (aiaj)j (I i ), is linearly independent for some i I. ⇥ −{ } ⇥ E2 E2 a0 a3 a2 a1 a0 a1 a2 Not A. I. Not A. I. 3 Triangulations In En, the largest number of affinely independent points is n + 1. Let a0,...,ad be any d + 1 affinely independent points in En. E2 E2 E2 a1 a0 a0 a1 a2 a0 4 Triangulations The simplex s spanned by the points a0,...,ad is the convex hull, conv( a ,...,a ), of these points, and is denoted by { 0 d} [a0,...,ad]. The points a0,...,ad are the vertices of s. E2 E2 E2 a1 a0 a0 a1 a2 a0 5 Triangulations The simplex s spanned by the points a0,...,ad is the convex hull, conv( a ,...,a ), of these points, and is denoted by { 0 d} [a0,...,ad]. The dimension, dim(s), of s is d, and s is called a d-simplex. E2 E2 E2 a1 a0 a0 a1 a2 a0 6 Triangulations The simplex s spanned by the points a0,...,ad is the convex hull, conv( a ,...,a ), of these points, and is denoted by { 0 d} [a0,...,ad]. In En, we have simplices of dimension 0, 1, . , n only. E2 E2 E2 a1 a0 a0 a1 a2 a0 7 Triangulations A 0-simplex is a point, a 1-simplex is a line segment, a 2- simplex is a triangle, and a 3-simplex is a tetrahedron, and so on. 0-simplex 1-simplex 2-simplex 3-simplex 8 Triangulations The convex hull of any nonempty (proper) subset of ver- tices of a simplex s is also a simplex, called a (proper) face of s. a2 a0 a1 a2 a2 a2 a0 a1 a0 a1 a0 a1 a 2-face: [a0, a1, a2] 3 proper 1-faces: [a0, a1], [a0, a2], [a1, a2] 3 proper 0-faces: [a0], [a1], [a2] 9 Triangulations A simplicial complex, , in En is a finite set of simplices in K En such that (1) if s and t s then t , and ⇥ K ⇥ K (2) if s t = ∆ then s t s, t, for all s, t , ⌅ ⇤ ⌅ ⇥ K where a b denotes "a is a (not necessarily proper) face of b". 10 Triangulations violates (1) violates (2) A simplicial complex 11 Triangulations The dimension, dim( ), of a simplicial complex is the K K largest dimension of a simplex in . We refer to a d- K dimensional simplicial complex as simply a d-(simplicial) complex. a 2-complex 12 Triangulations Note that a simplicial complex is a discrete object (i.e., a finite collection of simplices). In turn, each simplex is a set of points. a 2-complex 13 Triangulations The (point) set consisting of the union of all points in the simplices of a simplicial complex, , is called the underly- K ing space of and denoted by . Note that is a contin- K |K| K uous object. its underlying space 14 Triangulations A triangulation of a nonempty and finite set, P, of points of En, is a simplicial complex, (P), such that all vertices of T (P) are in P and the union of all simplices of (P) equals T T conv(P). E2 5 points in E2 convex hull of the 5 points 15 Triangulations A triangulation of a nonempty and finite set, P, of points of En, is a simplicial complex, (P), such that all vertices of T (P) are in P and the union of all simplices of (P) equals T T conv(P). E2 two triangulations of the same point set 16 Triangulations In our definition we do assume that the affine hull of P has dimension n. So, a triangulation of P may contain no simplex of dimension n, such as the example below for n = 2: P conv(P) (P) T E2 17 Triangulations Note that not all points of P need to be vertices of (P), ex- T cept for the extremes points of conv(P), which are always in (P). T Whenever all points in P are vertices of (P), we call (P) T T a full-triangulation. This is the type of triangulation we will study. From now on, we drop the word "full" and refer to the term "triangulation" as a full-triangulation of the given point set. 18 Triangulations Theorem 3.1. Every nonempty and finite set, P E2, ad- ⇢ mits a triangulation, which partitions the convex hull of P. (proof discussed in the end of the lecture) 19 The Delaunay Triangulation Let P be a nonempty and finite set of points in E2. For the time being, let us assume that (1) not all points of P are collinear, and (2) no four points of P lie in the circumference defined by 3 of them. Observe that "(1) ) P 3". | | ≥ E2 E2 co-circularity collinearity 20 The Delaunay Triangulation The Lifting Procedure Let w : E2 R be the function defined as ! w(p)=x2 + y2 , for every p =(x, y) E2. 2 21 The Delaunay Triangulation Note that w can be seen as a height function that lifts the point p =(x, y) to the paraboloid of equation z = x2 + y2 in E3. E3 w(p) p =(x, y) 22 The Delaunay Triangulation Let Pw = (x, y, w(x, y)) E3 (x, y) P . { ∈ | ∈ } E3 w(p) (x, y, w(x, y)) p =(x, y) 23 The Delaunay Triangulation Note that P is the orthogonal projection of Pw onto the xy- plane. E3 w(p) (x, y, w(x, y)) p =(x, y) 24 The Delaunay Triangulation Consider the convex hull, conv(Pw), of Pw. Denote it by . P E3 25 The Delaunay Triangulation Consider the convex hull, conv(Pw), of Pw. Denote it by . P P E3 26 The Delaunay Triangulation Project the lower envelope of onto the xy-plane. P orthogonally P E3 27 The Delaunay Triangulation Project the lower envelope of onto the xy-plane. P P E3 28 The Delaunay Triangulation If the result of the projection of the lower envelope of is a P triangulation, then we call it the Delaunay triangulation of P. We denote the Delaunay triangulation of P by (P). DT E2 29 The Delaunay Triangulation Proposition 3.2. Let S E3 be the paraboloid given by ⇢ the equation z = x2 + y2, and let H E3 be a non-vertical ⇢ hyperplane, i.e., one whose normal vector has non-zero last coordinate. Let C be the projection of H S into E2 \ obtained by dropping the last coordinate of all points in H S. Then, C is either empty, a single point, or a circum- \ ference. 30 The Delaunay Triangulation Proposition 3.2. z S H y C x (proof on the board) 31 The Delaunay Triangulation Lemma 3.3. Let Q P be any subset of P with 3 affinely ⇢ independent points. Then, Qw corresponds to the vertex set of a lower facet of the polytope, , if and only if all P points in Q lie on a circle and all points of P Q are outside − it. 32 The Delaunay Triangulation Lemma 3.3. z S H y C x (proof on the board) 33 The Delaunay Triangulation By hypothesis, no four points of P lie in the same circum- ference. So, Lemma 3.3 implies that all facets of the lower envelope of are triangles in E3, and so are their projections onto P E2. By hypothesis, not all points of P are collinear. This means that the proper faces of the lower envelope of are trian- P gles. 34 The Delaunay Triangulation What can we conclude from the previous remarks? The projection of the lower envelope of is a set of trian- P gles. It turns out that — with a little bit of an effort — we can also show that this set of triangles, along with their edges and vertices, is a triangulation of P. This implies that the edges and vertices of the triangles must be in the lower envelope of . P 35 The Delaunay Triangulation By definition, the triangulation resulting from the projec- tion of the lower envelope is the Delaunay triangulation, (P). DT 36 The Delaunay Triangulation Lemma 3.4. Let P be a nonempty and finite set of points of E2. If no four points of P lie in the same circumference and not all points of P lie in the same line, then the Delau- nay triangulation, (P), of P exists. Furthermore, it is DT unique. (proof on the board) 37 The Delaunay Triangulation What if one these two assumptions does not hold? E2 E2 co-circularity collinearity 38 The Delaunay Triangulation If all points of P are collinear, then is 2-dimensional P and its lower envelope is a polygonal chain containing all points of P. E3 39 The Delaunay Triangulation The projection of the lower envelope is also a polygonal chain containing all points of P, which is a triangulation as well.