Geometry

Floating point math Avoid floating point when you can –If you are given a fixed number of digits after the decimal, multiply & use integers Be careful when you can’t –Create EPSILON, “small enough” distance based on precision –Two numbers are equal if |x-y| < EPSILON

Lines Any 2 points determine a line –Four coordinates in 2d: x1,y1, x2, y2 –(y-y1)/(x-x1) = (y2-y1)/(x2-x1) Slope-intercept form y= mx + b (except vertical line) M is slope (tangent of line’s angle)

General Form: ax+by+c=0 Works for all 2d lines, vertical & horizontal Need canonical form –Set b=1 or b=0 (textbook) –Set a*a+b*b =1 (more typical in mathematics) –If a*a+b*b=1, then a=sin(theta), b=cos(theta)

Angle between lines Angle of line with horizontal: Atan(m) Angle between 2 lines (slope-int) atan(m1)-atan(m2) (or that) Or: atan((m2-m1)/(1+m1m2)) Angle between 2 lines (ax+by+c) Atan((a1b2-a2b1) / (a1a2+b1b2))

Intersection of Lines Directly (formula on p. 293) By search –Given 2 lines in “general position” –The intersection is the point at which y2 switches from being above y1 to below y1 –Binary search x values to find this point. Special cases: lines with same slope –Same line if they share a point –Parallel otherwise

Perpendicular Lines Slope m vs. slope -1/m Good for finding “closest point” Slope = m Slope = -1/m (right angle, 90 degrees) Point p Closest point on L1 is intersection with perp line through p

Voronoi Diagram Start with a set of points Compute the perpendicular bisectors of the line segments between the points These bisectors form the boundaries of regions around each point A test point in a point’s region is closer to that point than any other point in the set Models cell towers and wireless base stations.

Triangles 3 angles add up to 180 degrees Right triangle has one right angle –Pythagorean Theorem (a*a+b*b=c*c) –Trig: sin = a/c, cos=b/c (a is opposite), tan = sin/cos or a/b a b c Theta (sin is a/c)

General Triangle Laws Law of sines –a/sin A = b/sin B = c/sin C Law of cosines –a*a = b*b+c*c - 2bc cosA –(If A = 90 degrees, last term is 0)

Triangle Inequality If a, b and c are sides of a triangle, a+b > c If a+b=c, then the 3 vertices are collinear This is a useful test for whether 3 points are collinear: dist(p1,p2) + dist(p2,p3) = dist(p1,p3) Dist(p1,p2) = sqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2))

Area of Polygon Given the coordinates of any polygon, the area is computed by the formula Total=0; For(int v=0;v<N;v++) Total += (x[v]-x[v-1])*(y[v]+y[v-1]); Special case for triangles on page 297

Circles Circle is locus of all points equidistant (radius) from a single point (center) Area = pi * r*r Circumference = 2*pi*r Pi = (and many more digits) Tangent: line touches circle at one point (perpendicular to radius at that point) Intersections: 0, 1 or 2 of 2 circles

Triangulation Knowing distance of 3 points from a query point Construct 3 circles of appropriate radius These circles will (approximately) intersect in one point This point is the query point