Foot to the Pedal: Constant Pedal Curves and Surfaces Andrew Fabian Hieu D. Nguyen Rowan University Joint Math Meetings January 13, 010
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. http://mathworld.wolfram.com/pedalcurve.html http://en.wikipedia.org/wiki/pedal_curve
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
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. 300-31; Opera Omnia: Series 1, Volume, pp. 14 36 (Available at The Euler Archive: http://www.math.dartmouth.edu/~euler/ ) English translation by Andrew Fabian (007). (Available at The Euler Archive: http://www.math.dartmouth.edu/~euler/ ) 4
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
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
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
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
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
Solution by Differential Geometry Curves with constant pedal ps () = cs () ins () = a Differentiate: c'( s) in( s) + c( s) in'( s) = 0 0 + 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
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 + + 1 x y 11
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 + + 1 x y z z z z z x y = a 1+ + x y x y 1
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
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
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
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
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
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
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. 300-31; also published in Opera Omnia: Series 1, Volume, pp. 14 36. Posted on the Euler Archive: http://www.math.dartmouth.edu/~euler/ 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: http://www.maa.org/editorial/euler/hedi%060%0curves%0and%0paradox.pdf 19