Foot to the Pedal: Constant Pedal Curves and Surfaces

Transcription

1 Foot to the Pedal: Constant Pedal Curves and Surfaces Andrew Fabian Hieu D. Nguyen Rowan University Joint Math Meetings January 13, 010

2 Pedal Curves The pedal of a curve c with respect to a point O (origin) is the locus of the foot of the perpendicular from O to the tangent of the curve.

3 Formula for Pedal Curves ct () = ( xt (), yt ()) Tangent vector: Normal vector: c'( t) = ( x'( t), y'( t)) nt () = ( y'(), t x'()) t [ x'( t)] + [ y'( t)] Let us denote the pedal of c by p. Then p is given by the formula pt () = ( ct () int ()) nt () [ xt ( ) y'( t) x'( t) yt ( )] y'( t) [ x'( t) yt ( ) xt ( ) y'( t)] x'( t) [ x'()] t + [ y'()] t [ x'()] t + [ y'()] t =, Inverse Problem (much more difficult): ct () pt () 3

4 Leonard Euler s E36 Paper Exposition de quelques paradoxes dans le calcul integral (Explanation of Certain Paradoxes in Integral Calculus) Originally published in Memoires de l'academie des sciences de Berlin 1, 1758, pp ; Opera Omnia: Series 1, Volume, pp (Available at The Euler Archive: ) English translation by Andrew Fabian (007). (Available at The Euler Archive: ) 4

5 Curves With Constant Pedal (E36) Determine a curve c whose pedal has constant distance a from the origin pt () xt () y'() t + ytx () '() t = = [ x'( t)] + [ y'( t)] a Euler s Differential Equation: ytx () '() t xt () y'() t = a [ x'()] t + [ y'()] t dy dy y x = a 1+ dx dx y xp = a 1+ p p = dy dx 5

6 Integrating by Differentiating (Euler) y xp = a 1+ p p = dy dx Differentiate: dp Case I: 0 dx x = dy dx ap 1+ p dp 1 p+ x = dx ap 1+ dp ap dp x = dx 1+ p dx y a 1 p xp p = + + = dp dx a 1+ p 6

7 Eliminate the parameter: x + y Solution to Case I: ap a = + 1+ p 1+ p a p = 1+ + a p = a x + y = a (c = p is a circle) 7

8 dp Case II: 0 dx = Thus p = constant = n y xp= a 1+ p y = nx+ a 1+ n (c is a tangent line; p is a point ) 8

9 Differential Geometry of Plane Curves Arc length parameter: () t s t = c '() t dt T() s = c'() s (unit tangent vector) T'( s) Nt () = (unit normal vector) T'( s) T N t 0 (arc length) κ = T'( s) (curvature) Frenet formulas: T'( s) = κ N( s) N'( s) = κt( s) 9

10 Solution by Differential Geometry Curves with constant pedal ps () = cs () ins () = a Differentiate: c'( s) in( s) + c( s) in'( s) = cs ( ) in'( s) = 0 cs ()( i κts ()) = 0 κcs () ic'() s = 0 ( T N) Case I: κ = 0 cs ( ) is a line Case II: κ 0 cs ( ) ic'( s) = 0 cs () ics () = k cs () = k= a (constant) (circle) 10

11 Pedal Surfaces The pedal of a surface M with respect to a point O (origin) is the locus of the foot of the perpendicular from O to the tangent plane of the surface. Let us denote the pedal of M by P. If M is described by z = f(x,y), then P is given by the formula n = pxy (, ) = ( zinn ) where n is the unit normal vector z z,,1 x y z z x y 11

12 Surfaces With Constant Pedal Determine a surface M whose pedal has constant distance k from the origin We will call a surface M with constant pedal a tangentially equidistant (TED) surface pxy (, ) = k= Thus our surface S satisfies z z x y + z x y z z x y z z z z z x y = a 1+ + x y x y 1

13 Differential Geometry of TED Surfaces TED surfaces (with constant pedal) Differentiate: M = x( u, v): D x( uv, ) in( uv, ) = k 3 xuin+ xn i u = 0+ xn i u = 0 xin+ xn i = 0+ xn i = 0 v v v Case I: { n, n } are linearly independent on M u v xx i u = 0 span{ nu, nv} = TM p xix= k(circle) xx i v = 0 Case II: { n, n } are NOT linearly independent on M u v Gauss curvature K = 0 on M (developable) M is a ruled surface 13

14 Ruled TED Surfaces x : D M 3 x( uv, ) = β( u) + vδ( u) β ( u) δ ( u) - directrix - ruling Cone Observation: Let M be a ruled surface whose directrix β is a curve that lies on the sphere S (k) of radius k. Then M is a TED surface with constant pedal β if and only if the tangent planes of M and S (k) agree on β. x x u v ( uv, ) = β '( u) + vδ '( u) ( uv, ) = δ ( u) 14

15 Lemma: Assume β( u) S ( k) and δ( u) Tβ ( u) S ( k). Then Tx uvm = Tβ us ( k) if and only if β '( u), δ( u), δ '( u) (, ) ( ) are coplanar. Theorem: Let M be a ruled surface having a coordinate patch of the form x( uv, ) = β( u) + vδ( u) where β( ) ( ). If β '( ), δ( ), δ '( ) are coplanar, then u S k u u u M is a developable ruled TED surface with constant pedal β. Explicit Construction of TED surfaces: Lemma: If β is a regular spherical curve and then βδ,, and δ' are coplanar. ( u) = ( u) '( u) Tβ ( u) S ( k), δ β β 15

16 Example 1: (Equator) β ( u) = (cos u,sin u,0) δ( u) = β( u) β '( u) = (0,0,1) x( uv, ) = β( u) + vδ( u) = (cos u,sin uv, ) Example : (Latitude circle) 1 β ( u) = (cos u,sin u,1) 1 δ( u) = β( u) β '( u) = ( cos u, sin u,1) 1 x( uv, ) = β( u) + vδ( u) = (( v)cos u,( v)sin u, + v) 16

17 Example 3: (Figure-8) β u u u u u ( ) = (cos sin,sin,cos ) 1 δ = β β = + x( uv, ) = β( u) + vδ( u) 3 ( u) ( u) '( u) ( (3 cos u)sin u,cos u,sin u) 1 = ( ( cos u+ v(3 + cos u))sin u, 3 vcos u+ sin u, cosu+ vsin u) 17

18 Further Research - Description of TED surfaces as the union a region R of the sphere and the developable ruled surface corresponding to the boundary of R - Must every (smooth) TED surface with constant pedal be either the sphere, a tangent plane, a developable ruled surface, or a union of a region of the sphere and a developable rule surface? - Generalization to n-dimensional TED manifolds 18

19 References A. Fabian, English Translation of Leonard Euler s E36 publication, Exposition de quelques paradoxes dans le calcul integral (Explanation of Certain Paradoxes in Integral Calculus), originally published in Memoires de l'academie des sciences de Berlin 1, 1758, pp ; also published in Opera Omnia: Series 1, Volume, pp Posted on the Euler Archive: A. Fabian and H. D. Nguyen, Paradoxical Euler: Integrating by Differentiating, Preprint (008). E. Sandifer, Curves and paradox, How Euler Did It (October 008 column article), MAA Online: 19