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21.3 Triangles in Two and Three Dimensions using none toinsert none with web,windows creating barcode 21.3 Triangles in Two and Three Dimensions USPS POSTNET Barcode Not since the time of Euclid has the lowly triangle attracted as much attention as it does today in computer graphics. Triangles and triangulation (the decomposition, or approximation, of complicated geometrical objects using only triangles) are at the heart of practically every computer-generated image. Three points, call them a, b, c, de ne a triangle.

They are its vertices. If the points are two-dimensional, the triangle lies in the two-dimensional plane. If the points have higher dimensionality, then the triangle oats in the corresponding Ddimensional space (most commonly D D 3).

For now, consider only the former case, with D D 2, so that a has coordinates .a0 ; a1 /, and similarly for b and c. Area.

The area of the triangle 4abc can be written in a number of equivalent ways, including a0 a1 1 D b0 b1 1 c0 c1 1 D .

b D .c D .a a/ b/ c/ .

c .a .b a/ D .

b0 b/ D .c0 c/ D .a0 a0 /.

c1 b0 /.a1 c0 /.b1 a1 / b1 / c1 / .

b1 .c1 .a1 a1 /.

c0 b1 /.a0 c1 /.b0 a0 / b0 / c0 / (21.

3.1) Here denotes the vector cross product, de ned in two dimensions simply by A B D A0 B1 B1 A0 (two dimensions only) (21.3.

2). Below, when we consider trian none for none gles in three dimensions, it will be the vector cross product forms in equation (21.3.1) that give a generalized formula for the area.

Let us also note in passing that the formulas for area are separately linear in each of the six coordinates a0 , a1 , b0 , b1 , c0 , and c1 . Equation (21.3.

1) can yield a value that is positive, zero, or negative: The area is a signed area. By convention (embodied in equation 21.3.

1), the area is positive if a traversal from a to b to c goes counterclockwise (CCW) around the triangle, and negative if it goes clockwise (CW). The area is zero if and only if the three points are collinear, in which case the triangle is degenerate. (In the formulas that follow, we will generally assume the nondegenerate case.

) The absolute value jAj is the (unsigned) area of the triangle in the conventional geometrical sense. It can also be calculated directly from the side lengths dab , dbc , and dca as follows: p (21.3.

3) jAj D s.s dab /.s dbc /.

s dca / where s is half the perimeter, s 1 .dab C dbc C dca / 2 (21.3.

4). (Does it go without saying th at you compute the side lengths by taking the coordinate differences and using the Pythagorean theorem ). 21. Computational Geometry u v u u w w v v u w w Figure 21.3.1.

Three kinds of triangle centers. (a) Incircle and incenter; bisectors of the three vertex angles meet at the incenter. (b) Circumcircle and circumcenter; perpendicular bisectors of the edges meet at the circumcenter.

(c) Centroid; lines from the edge midpoints to the opposite vertices meet at the centroid.. Related Circles. Every nondeg none for none enerate triangle has an inscribed circle or incircle, which is the largest circle that can be drawn inside the triangle. The incircle is tangent to all three sides of the triangle.

Lines from its center, the incenter, to each vertex bisect the angle at that vertex (see Figure 21.3.1).

If q is the incenter point, with coordinates .q0 ; q1 /, then its location is given by qi D 1 .dbc ai C dca bi C dab ci / 2s .

i D 0; 1/ (21.3.5).

while its radius is given by rin D .s dab /.s dbc /.

s s dca / 1=2 (21.3.6).

Every nondegenerate triangle none none also has a circumscribed circle or circumcircle, which is the unique circle that goes through its three vertices. Suppose Q is the circumcenter point, with coordinates .Q0 ; Q1 /.

Let ba 0 and ba 1 denote the coordinate differences b0 a0 and b1 a1 , respectively; and similarly for ca 0 and ca 1 . Then, in 2 2 determinant form, ba 0 ba 1 1 . ba 0 /2 C .

ba 1 /2 ba 1 Q0 D a0 C ca 0 ca 1 2 . ca 0 /2 C . ca 1 /2 ca 1 (21.

3.7) ba 0 ba 1 1 ba 0 . ba 0 /2 C .

ba 1 /2 Q1 D a1 C ca 0 ca 1 2 ca 0 . ca 0 /2 C . ca 1 /2 The circumcenter is, by de nition, the same distance from all three vertices.

Therefore the radius of the circumcircle is p (21.3.8) rcircum D .

Q0 a0 /2 C .Q1 a1 /2 where Q0 and Q1 are given above. (Obviously you can save the semi- nal results in equation 21.

3.7 for this computation, before adding a0 or a1 .) Later, in 21.

6, we will be calculating a lot of circumcircles. We use the following simple de nition of a structure Circle, and a routine circumcircle() that directly implements equations (21.3.

7) and (21.3.8).

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