Basic tool: orientation tests

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Basic tool: orientation tests Vera Sacristán Computational Geometry Facultat d Informàtica de Barcelona Universitat Politècnica de Catalunya

Orientation test in R 2 Given 3 points p,, r in the plane, efficiently and robustely decide whether r lies to the left, to the right or on the oriented line p. r 1 r 3 p r 2

Orientation test in R 2 Given 3 points p,, r in the plane, efficiently and robustely decide whether r lies to the left, to the right or on the oriented line p. r 1 Consider the vectors p... p r 2 r 3

Orientation test in R 2 Given 3 points p,, r in the plane, efficiently and robustely decide whether r lies to the left, to the right or on the oriented line p. r 1 Consider the vectors p and pr. p r 2 r 3

Orientation test in R 2 Given 3 points p,, r in the plane, efficiently and robustely decide whether r lies to the left, to the right or on the oriented line p. r 1 Consider the vectors p and pr. p Point r lies on the line p x p x r x p x y p y r y p y = 0 Point r lies to the left of the oriented line p x p x y p y Point r lies to the right of the oriented line p x p x y p y r x p x r y p y r x p x r y p y > 0 r 3 r 2 < 0

Orientation test in R 2 Given 3 points p,, r in the plane, efficiently and robustely decide whether r lies to the left, to the right or on the oriented line p. Consider the vectors p and pr. p Point r lies on the line p x p x r x p x y p y r y p y = 0 Point r lies to the left of the oriented line p x p x r x p x y p y r y p y > 0 Point r lies to the right of the oriented line p x p x r x p x y p y r y p y < 0 Notation det(p,, r) = x p x r x p x y p y r y p y = p x x r x p y y r y 1 1 1 r 1 r 2 r 3

Relative position point - line l: a line (through points p and ) r: a point Output: Relative position of r w.r.t. l. p r l

Relative position point - line l: a line (through points p and ) r: a point Output: Relative position of r w.r.t. l. p r l Intersection test line segment - line l: a line (through points p and ) s: a line segment (with endpoints a and b) Output: a Yes/No they intersect b p a b b l a

Intersection test line segment - halfline l: halfline (from p through ) s: a line segment (with endpoints a and b) Output: Yes/No they intersect b p a b a b l a

Intersection test line segment - halfline l: halfline (from p through ) s: a line segment (with endpoints a and b) Output: Yes/No they intersect b p a b a b l a Intersection test line segment - line segment s 1 : a line segment (with endpoints p and ) s 2 : a line segment (with endpoints a and b) Output: Yes/No they intersect p a b a a b b

Intersection test line - polygon l: a line (through p and ) P : a polygon (with vertices p 1, p 2,..., p m ) Yes/No they intersect. If they do, the edges of P intersecting l

Intersection test line - polygon l: a line (through p and ) P : a polygon (with vertices p 1, p 2,..., p m ) Yes/No they intersect. If they do, the edges of P intersecting l What if the polygon is convex?

Turn orientation A polygonal line p 1, p 2,..., p n Output: Left/right classification of its turns p 1

Turn orientation A polygonal line p 1, p 2,..., p n Output: Left/right classification of its turns p 1 Point in triangle test A triangle T with vertices p 1, p 2, p 3 A uery point Output: Relative position of w.r.t. T T

Point in polygon test A polygon p 1, p 2,..., p n A uery point Output: Yes/No P

Point in polygon test A polygon p 1, p 2,..., p n A uery point Output: Yes/No P What if the polygon is convex?

Supporting lines point - polygon A polygon P with vertices p 1, p 2,..., p n A point not belonging to the convex hull of P Output: Lines through and P that leave all of P to one side P

Supporting lines point - polygon A polygon P with vertices p 1, p 2,..., p n A point not belonging to the convex hull of P Output: Lines through and P that leave all of P to one side P

Supporting lines point - polygon A polygon P with vertices p 1, p 2,..., p n A point not belonging to the convex hull of P Output: Lines through and P that leave all of P to one side What if the polygon is convex? P

What happens in R 3?

Oriented area of a triangle p r Oriented area (p,, r) = 1 2 p x x r x p y y r y 1 1 1 Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

Oriented area of a triangle p r Oriented area (p,, r) = 1 2 p x x r x p y y r y 1 1 1 Oriented volume of a tetrahedron a d c b Oriented volume (a, b, c, d) = 1 6 a x b x c x d x a y b y c y d y a z b z c z d z 1 1 1 1 Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

Relative position point-plane If a, b, and c are not aligned, they define a plane π. c π a b Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

Relative position point-plane If a, b, and c are not aligned, they define a plane π. π a b c Point x lies in the halfspace π + det(x, a, b, c) > 0. Point x lies in the plane π det(x, a, b, c) = 0. Point x lies in the halfspace π det(x, a, b, c) < 0. Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

2D application: Relative position point-circle Proposition 1 The intersection of the paraboloid whose euation is z = x 2 + y 2 with a non vertical plane is a curve that projects orthogonally onto a circle in the plane z = 0. Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

2D application: Relative position point-circle Proposition 1 The intersection of the paraboloid whose euation is z = x 2 + y 2 with a non vertical plane is a curve that projects orthogonally onto a circle in the plane z = 0. Proof: } } } z = x 2 + y 2 x 2 + y 2 = 2ax + 2by + c (x a) 2 + (y b) 2 = c + a 2 + b 2 z = 2ax + 2by + c z = 0 z = 0 If c < a 2 b 2, the intersection is empty. If c = a 2 b 2, the intersection is the point (a, b, a 2 + b 2 ). If c > a 2 b 2, the intersection is a curve (in fact, an ellipse) that projects onto the circle of the plane z = 0 whose center is (a, b) and whose radius is r = c + a 2 + b 2. Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

2D application: Relative position point-circle Proposition 2 Let x, a, b, c be four points in the plane z = 0, and let x, a, b, c respectively be their vertical projections onto the paraboloid z = x 2 + y 2. If a, b, c are not aligned, let C be the circle through a, b, c, and let π be the plane through a, b, c. Then: The point x lies in the circle C if and only if x lies in the plane π. The point x lies in the interior of the circle C if and only if x lies in the lower half-space determined by π. The point x lies in the exterior of the circle C if and only if x lies in the upper half-space determined by π. Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

2D application: Relative position point-circle Proposition 2 Let x, a, b, c be four points in the plane z = 0, and let x, a, b, c respectively be their vertical projections onto the paraboloid z = x 2 + y 2. If a, b, c are not aligned, let C be the circle through a, b, c, and let π be the plane through a, b, c. Then: The point x lies in the circle C if and only if x lies in the plane π. The point x lies in the interior of the circle C if and only if x lies in the lower half-space determined by π. The point x lies in the exterior of the circle C if and only if x lies in the upper half-space determined by π. Proof: Due to Proposition 1 and the convexity of the paraboloid. Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

2D application: Relative position point-circle Corollary Let a, b and c be three non aligned points in the plane, that appear angularly sorted in counterclockwise order in the circle C that they determine. Let x be any point in the plane. Then: The point x lies in the circle C if and only if det(x, a, b, c ) = 0. The point x lies in the interior of C if and only if det(x, a, b, c ) < 0. The point x lies in the exterior of C if and only if det(x, a, b, c ) > 0. Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

2D application: Relative position point-circle Observation In order to compute the determinant of the previous corollary, it is convenient to do the calculations in terms of the differences between the values of the coordinates of the points involved, and to avoid making calculations (specially, products) in terms of the coordinate values, if possible: x 1 x 2 x 2 1 + x 2 2 1 a 1 a 2 a 2 1 + a 2 2 1 b 1 b 2 b 2 1 + b 2 2 1 c 1 c 2 c 2 1 + c 2 2 1 = b 1 a 1 b 2 a 2 (b 1 a 1 )(b 1 + a 1 ) + (b 2 a 2 )(b 2 + a 2 ) c 1 a 1 c 2 a 2 (c 1 a 1 )(c 1 + a 1 ) + (c 2 a 2 )(c 2 + a 2 ) x 1 a 1 x 2 a 2 (x 1 a 1 )(x 1 + a 1 ) + (x 2 a 2 )(x 2 + a 2 ) Discrete and Algorithmic Geometry, Facultat de Matemàtiues i Estadística, UPC

A BASIC TOOL: ORIENTATION TESTS FURTHER READING J. O Rourke Computational Geometry in C Cambridge University Press, 1994 (2nd ed. 1998), pp. 17-35. F. P. Preparata and M. I. Shamos Computational Geometry: An Introduction Springer-Verlag, 1985, pp. 36-45.

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