Practical Linear Algebra: A Geometry Toolbox

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1 Practical Linear Algebra: A Geometry Toolbox Third edition Chapter 3: Lining Up: 2D Lines Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book c 2013 Farin & Hansford Practical Linear Algebra 1 / 27

2 Outline 1 Introduction to 2D Lines 2 Defining a Line 3 Parametric Equation of a Line 4 Implicit Equation of a Line 5 Explicit Equation of a Line 6 Converting Between Parametric and Implicit Equations 7 Distance of a Point to a Line 8 The Foot of a Point 9 A Meeting Place: Computing Intersections 10 WYSK Farin & Hansford Practical Linear Algebra 2 / 27

3 Introduction to 2D Lines 2D lines are the building blocks for many geometric constructions Two sets of parallel lines overlaid Interference pattern called Moiré pattern Used in optics for checking the properties of lenses Chapter focus: representations for lines distance from a line intersections Farin & Hansford Practical Linear Algebra 3 / 27

4 Defining a Line Two elements of 2D geometry define a line: Elements can be two points a point and a vector parallel to the line a point and a vector perpendicular to the line Normal to a line: unit vector that is perpendicular (or orthogonal) to a line Farin & Hansford Practical Linear Algebra 4 / 27

5 Defining a Line Families of lines One family of lines shares a common point The other family of lines shares the same normal Farin & Hansford Practical Linear Algebra 5 / 27

6 Parametric Equation of a Line l(t) = p+tv where p E 2 and v R 2 Scalar value t is the parameter Evaluating for specific parameter generates a point on the line Farin & Hansford Practical Linear Algebra 6 / 27

7 Parametric Equation of a Line Parametric form l(t) = p+tv in terms of barycentric coordinates: Let v = q p, then l(t) = (1 t)p+tq (1 t) and t are the barycentric coordinates of a point on the line with respect to p and q, resp. Typically referred to as linear interpolation Farin & Hansford Practical Linear Algebra 7 / 27

8 Parametric Equation of a Line Barycentric combination: sum of coefficients of points is one l(t) = (1 t)p+tq For t [0,1]: generates points on line between p and q This is a convex combination t < 0 generates points on the line behind p t > 1 generates points past q (Interpret as scaling of v in l(t) = p+tv) This is extrapolation Farin & Hansford Practical Linear Algebra 8 / 27

9 Parametric Equation of a Line Parametric form very good for computing points on a line Example: compute ten equally spaced points on line segment through p and q l(t) = (1 t)p+tq t = i/9, i = 0,...,9 i = 0 t = 0 p i = 9 t = 1 q Equally spaced parameter values correspond to equally spaced points Parametrization: speed line is traversed Farin & Hansford Practical Linear Algebra 9 / 27

10 Implicit Equation of a Line Point p and a vector a perpendicular to the line For any point x on the line a (x p) = 0 a and (x p) are perpendicular If a unit length, called point normal form Expand: a 1 x 1 +a 2 x 2 +( a 1 p 1 a 2 p 2 ) = 0 Results in familiar form: ax 1 +bx 2 +c = 0 Farin & Hansford Practical Linear Algebra 10 / 27

11 Implicit Equation of a Line Given: two points on the line [ ] [ ] 2 6 p = and q = 2 4 Find: implicit line ax 1 +bx 2 +c = 0 [ ] 4 v = q p = 2 [ ] 2 a = 4 c = = 4 Implicit equation of the line: 2x 1 +4x 2 4 = 0 (Not point normal form) Farin & Hansford Practical Linear Algebra 11 / 27

12 Implicit Equation of a Line Implicit form good for deciding if an arbitrary point lies on the line For given point x f = ax 1 +bx 2 +c If f = 0 then the point is on the line Farin & Hansford Practical Linear Algebra 12 / 27

13 Implicit Equation of a Line Numerical caveat: Checking equality f = 0.0 with floating point numbers not recommended Tolerance ǫ needed Meaningful tolerance: true distance of x to line d = f a Sign of d indicates side of the line Farin & Hansford Practical Linear Algebra 13 / 27

14 Explicit Equation of a Line Explicit form is closely related to the implicit form ax 1 +bx 2 +c = 0 Expresses x 2 as a function of x 1 x 2 = a b x 1 c b Simpler: x 2 = âx 1 +ˆb Geometric meaning of coefficients: â is the slope rise/run ˆb is the e 2 -intercept Example: x 2 = 1/3x 1 +1 Drawback: vertical line has infinite slope Farin & Hansford Practical Linear Algebra 14 / 27

15 Converting Between Parametric and Implicit Equations Advantages to both the parametric and implicit representations of a line Depending on the geometric algorithm may be convenient to use one form rather than the other Ignore the explicit form: not very useful for general 2D geometry Farin & Hansford Practical Linear Algebra 15 / 27

16 Converting Between Parametric and Implicit Equations Parametric to Implicit Given: l(t) = p+tv Find: coefficients a,b,c of ax[ 1 +bx] 2 +c = 0 v2 Solution: Form a vector a = perpendicular to vector v Determines the coefficients a = a 1 and b = a 2 Use p from l(t) to compute c = (a 1 p 1 +a 2 p 2 ) v 1 Farin & Hansford Practical Linear Algebra 16 / 27

17 Converting Between Parametric and Implicit Equations Implicit to Parametric Given: ax 1 +bx 2 +c = 0 Find: l(t) = p+tv Solution: Need [ ] one point on the line and a vector parallel to the line b Vector v = is perpendicular to a a Point candidates: intersections with the e 1 - or e 2 -axis [ ] [ ] c/a 0 or 0 c/b For numerical stability, choose the intersection closest to the origin Farin & Hansford Practical Linear Algebra 17 / 27

18 Converting Between Parametric and Implicit Equations Non-uniqueness of representations Two parametric representations for the same line Lines will be traced will differently Conversion process for example parametric implicit parametric Original and final parametric form different in general Farin & Hansford Practical Linear Algebra 18 / 27

19 Distance of a Point to a Line Robot will travel along the line (thick black) Points represent objects in the room Verify needed robot clearance (gray) by computing distance of point to line Smallest distance d(r, l) of a point to a line is the orthogonal or perpendicular distance Farin & Hansford Practical Linear Algebra 19 / 27

20 Distance of a Point to a Line Starting with an Implicit Line Given: l : ax 1 +bx +c and point r Find: d(r, l) [ ] a Solution: let a = b d = ar 1 +br 2 +c a = a (r p) a Let w = r p v = a w = a w cos(θ) cos(θ) = d w then v = a d d = ar 1 +br 2 +c a Farin & Hansford Practical Linear Algebra 20 / 27

21 Distance of a Point to a Line Starting with a Parametric Line Given: l(t) = p+tv and a point r Find: d(r, l) Solution: Let w = r p d = w sin(α) sin(α) = 1 cos(α) 2 cos(α) = v w v w Farin & Hansford Practical Linear Algebra 21 / 27

22 The Foot of a Point Which point on the line is closest to a given point? This point is the foot of the given point Given: l(t) = p+tv and point r Find: q, the foot of r Solution: t such that q = p+tv Define w = r p Solve for t cos(θ) = tv w = v w v w t = v w v 2 Farin & Hansford Practical Linear Algebra 22 / 27

23 A Meeting Place: Computing Intersections Intersection problems important in many applications Fun figure: finding intersections to create an artistic image Possible questions: Do the lines intersect? At what point do they intersect? At what parameter value on one or both lines is the intersection point? Question(s) + line representation(s) best method Farin & Hansford Practical Linear Algebra 23 / 27

24 A Meeting Place: Computing Intersections Parametric and Implicit Given: l 1 (t) = p+tv l 2 : ax 1 +bx 2 +c = 0 Find: intersection point i Solution: i = p+ˆtv a[p 1 +ˆtv 1 ]+b[p 2 +ˆtv 2 ]+c = 0 One equation and one unknown ˆt = c ap 1 bp 2 av 1 +bv 2 i = l 1 (ˆt) What if the lines are parallel? Farin & Hansford Practical Linear Algebra 24 / 27

25 A Meeting Place: Computing Intersections Both Parametric Given: l 1 (t) = p+tv l 2 (s) = q+sw Find: intersection point i Solution: ˆt and ŝ such that Rewritten: p+ˆtv = q+ŝw. ˆtv ŝw = q p Two equations and two unknowns If v and w linearly dependent: no solution Geometric meaning? Farin & Hansford Practical Linear Algebra 25 / 27

26 A Meeting Place: Computing Intersections Both Implicit Given: l 1 : ax 1 +bx 2 +c = 0 l 2 : āx 1 + bx 2 + c = 0 Find: intersection point i = ˆx that satisfies l 1 and l 2 Solution: aˆx 1 +bˆx 2 = c āˆx 1 + bˆx 2 = c Two equations and two unknowns Lines parallel a and ā linearly dependent Farin & Hansford Practical Linear Algebra 26 / 27

27 WYSK parametric form of a line linear interpolation point normal form implicit form of a line explicit form of a line line through two points line defined by a point and a vector parallel to the line line defined by a point and a vector perpendicular to the line distance of a point to a line line form conversions foot of a point intersection of lines Farin & Hansford Practical Linear Algebra 27 / 27

Practical Linear Algebra: A Geometry Toolbox

Practical Linear Algebra: A Geometry Toolbox Third edition Chapter 1: Affine Maps in 3D Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/pla