Parametric Curves, Lines in Space
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1 Parametric Curves, Lines in Space Calculus III Josh Engwer TTU 02 September 2014 Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
2 PART I PART I: 2D PARAMETRIC CURVES Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
3 Parametric Curves in R 2 (Introduction) Definition A 2D parametric curve has the following form: x = f (t) y = g(t) t I, where I is an interval f, g C(I) Each point (x, y) on the curve depends on a parameter, t R. A particular choice of functions f, g and interval I is called a parameterization of the curve. ALTERNATIVE NOTATION: x(t) = f (t) y(t) = g(t) is used to emphasize that x and y are functions of t. t I ALTERNATIVE NOTATION: x = f (t), y = g(t), t I REMARK: The t I piece is omitted when t Dom(f ) Dom(g) Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
4 Parametric Curves in R 2 (Introduction) (DEMO) PARAMETRIC CURVES (Click below): Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
5 Parametric Curves (Parameterizations are not Unique) (DEMO) COMPARING PARAMETERIZATIONS (Click below): REMARK: Focus on choosing simplest parameterization (see next slide). Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
6 Parametric Curves in R 2 (Conversions) CONVERSION PROCEDURE Solve for t in one eqn and substitute into the other eqn. 2D Parametric Rectangular Restrict the ranges of x & y if necessary. x = t Explicit Rectangular 2D Parametric y = f (x) = y = f (t) t Dom(f ) x = g(t) Explicit Rectangular 2D Parametric x = g(y) = y = t Explicit Polar 2D Parametric r = f (θ) = t Dom(g) x = f (t) cos t y = f (t) sin t t Dom(f ) *** ALL OTHER CONVERSION POSSIBILITIES ARE TOO DIFFICULT *** Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
7 Parametric Curves in R 3 (Introduction) Definition A 3D parametric curve has the following form: x = f (t) y = g(t) z = h(t) t I, where I is an interval f, g, h C(I) Each point (x, y, z) on the curve depends on a parameter, t R. A particular choice of functions f, g, h and interval I is called a parameterization of the curve. REMARK: General 3D parametric curves will be treated in Chapter 10. For now, consider only 3D lines in space. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
8 PART II PART II: LINES IN SPACE Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
9 Lines in R 2 (Canonical Forms from Algebra) Recall from Algebra the following canonical forms of a line on the xy-plane: TYPE CANONICAL FORM REMARK(s) Point-Slope Form y y 0 = m(x x 0 ) m Slope of Line Line contains point (x 0, y 0 ) Slope-Intercept Form y = mx + b m Slope of Line b y-intercept : (0, b) Both-Intercepts Form x a + y m Slope of Line b = 1 a x-intercept : (a, 0) b y-intercept : (0, b) General Form Ax + By = C A, B, C R Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
10 Lines in R 2 (Canonical Forms from Algebra) Recall from Algebra the following canonical forms of a line on the xy-plane: TYPE CANONICAL FORM REMARK(s) Point-Slope Form y y 0 = m(x x 0 ) m Slope of Line Line contains point (x 0, y 0 ) Slope-Intercept Form y = mx + b m Slope of Line b y-intercept : (0, b) Both-Intercepts Form x a + y m Slope of Line b = 1 a x-intercept : (a, 0) b y-intercept : (0, b) General Form Ax + By = C A, B, C R Unfortunately, none of these forms are useful for lines in space. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
11 Lines in R 2 (Canonical Forms from Algebra) Recall from Algebra the following canonical forms of a line on the xy-plane: TYPE CANONICAL FORM REMARK(s) Point-Slope Form y y 0 = m(x x 0 ) m Slope of Line Line contains point (x 0, y 0 ) Slope-Intercept Form y = mx + b m Slope of Line b y-intercept : (0, b) Both-Intercepts Form x a + y m Slope of Line b = 1 a x-intercept : (a, 0) b y-intercept : (0, b) General Form Ax + By = C A, B, C R Unfortunately, none of these forms are useful for lines in space. The solution: Use Vectors! Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
12 Lines (Parametric Form Derivation) Given line l vector v = v 1, v 2 and containing point P(x 0, y 0 ) Let Q(x, y) be any point on line l. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
13 Lines (Parametric Form Derivation) Given line l vector v = v 1, v 2 and containing point P(x 0, y 0 ) Form vector PQ = x x 0, y y 0. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
14 Lines (Parametric Form Derivation) Given line l vector v = v 1, v 2 and containing point P(x 0, y 0 ) Notice that PQ v Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
15 Lines (Parametric Form Derivation) Given line l vector v = v 1, v 2 and containing point P(x 0, y 0 ) PQ v = x x 0, y y 0 = t v 1, v 2 Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
16 Lines (Parametric Form Derivation) Given line l vector v = v 1, v 2 and containing point P(x 0, y 0 ) PQ v = x x 0, y y 0 = t v 1, v 2 = x x 0, y y 0 = tv 1, tv 2 Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
17 Lines (Parametric Form Derivation) Given line l vector v = v 1, v 2 and containing point P(x 0, y 0 ) PQ v = x x 0, y y 0 = t v 1, v 2 = { x x 0, y y 0 = tv 1, tv 2 x x0 = tv Equate each component: 1 y y 0 = tv 2 Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
18 Lines (Parametric Form Derivation) Given line l vector v = v 1, v 2 and containing point P(x 0, y 0 ) PQ v = x x 0, y y 0 = t v 1, v 2 = { x x 0, y y 0 = tv 1, tv 2 x x0 = tv Equate each component: 1 { y y 0 = tv 2 x = x0 + tv Solve for (x, y): 1 y = y 0 + tv 2 Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
19 Lines in R 2 (Parametric Form) Definition The parametric form of line l containing point P 0 (x 0, y 0 ) and parallel to vector v = v 1, v 2 is: x = x 0 + v 1 t l : y = y 0 + v 2 t t R Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
20 Lines in R 3 (Parametric Form) Definition The parametric form of line l containing point P 0 (x 0, y 0, z 0 ) and parallel to vector v = v 1, v 2, v 3 is: x = x 0 + v 1 t y = y l : 0 + v 2 t z = z 0 + v 3 t t R Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
21 Lines in R 3 (Intersecting Lines) Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
22 Lines in R 3 (Parallel Lines) Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
23 Lines in R 3 (Coincident Lines) Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
24 Lines in R 3 (Skew Lines) The dashed line segment indicates that line l 2 is behind line l 1. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
25 Lines in R 3 (Classification Flowchart) Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
26 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
27 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
28 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
29 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
30 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
31 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. sin θ = d QP Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
32 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. d = QP sin θ Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
33 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. d v = v QP sin θ Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
34 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. d v = v QP Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
35 Lines in R 3 (Distance from Point to Line Derivation) Find the shortest distance from point P to line l. d = v QP v Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
36 Lines in R 3 (Distance from Point to Line) Theorem The shortest distance, d, from point P to line l is: d = v QP v where Q is any point on line l, and v is any vector parallel to line l. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
37 Fin Fin. Josh Engwer (TTU) Parametric Curves, Lines in Space 02 September / 37
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