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Geometry

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.

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Geometry

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  1. Geometry

  2. 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

  3. 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)

  4. 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)

  5. Angle between lines • Angle of line with horizontal: Atan(m) • Angle between 2 lines (slope-int) atan(m1)-atan(m2) (or 180 - that) Or: atan((m2-m1)/(1+m1m2)) • Angle between 2 lines (ax+by+c) Atan((a1b2-a2b1) / (a1a2+b1b2))

  6. 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

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

  8. 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.

  9. 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 c a Theta (sin is a/c) b

  10. 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)

  11. 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))

  12. 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

  13. Circles • Circle is locus of all points equidistant (radius) from a single point (center) • Area = pi * r*r • Circumference = 2*pi*r • Pi = 3.1415926 (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

  14. 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

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