Area-Invariant Pedal-Like Curves Derived from the Ellipse
aa r X i v : . [ m a t h . M G ] S e p AREA-INVARIANT PEDAL-LIKE CURVESDERIVED FROM THE ELLIPSE
DAN REZNIK, RONALDO GARCIA, AND HELLMUTH STACHEL
Abstract.
We study six pedal-like curves associated with the ellipse whichare area-invariant for pedal points lying on one of two shapes: (i) a circleconcentric with the ellipse, or (ii) the ellipse boundary itself. Case (i) is acorollary to properties of the Curvature Centroid (Krümmungs-Schwerpunkt)of a curve, proved by Steiner in 1825. For case (ii) we prove area invariancealgebraically. Explicit expressions for all invariant areas are also provided.
Keywords ellipse, pedal, contrapedal, evolute, curvature centroid, invariance.
MSC Introduction
Consider an ellipse E and a fixed point M . Let E n denote the negative-pedalcurve with respect to M [6], i.e., the envelope of lines L ( t ) through a point P ( t ) on E and perpendicular to P ( t ) − M ; see Figure 1. This article was motivated bya recent result [4]: E n is a three-cusp area-invariant deltoid for all M on E ; seeFigure 1 (top right).Let E p , E c denote the pedal , and contrapedal , curves of E with respect to a point M [6]; see Figures 2. Recall the contrapedal of a plane curve is the pedal of theevolute [11, Contrapedal]. For the ellipse, the evolute is a 4-cusp astroid [11, EllipseEvolute]; see Figure 3. Additionally, define: • The
Rotated Pedal Curve E θ , the locus of foot Q θ of a perpendicular droppedfrom M onto the line through P ( t ) oriented along a θ -rotated tangent tothe ellipse, Figure 4(left). • The
Interpolated Pedal Curve E µ , the locus a point Q µ = (1 − µ ) Q p + µQ c ( µ is a constant), i.e., an affine combination of pedal and contrapedal feet,Figure 4(right). • The
Hybrid Pedal Curve E ∗ , the locus of the intersection Q ∗ of L ( t ) withthe line from M to Q p , Figure 9. • The
Pseudo Talbot Curve E † , i.e., the Negative Pedal Curve of E ∗ , Fig-ure 10.Let A , A p , A c , A θ , A µ , A ∗ , and A † denote the areas of E , E p , E c , E θ , E µ , E ∗ , and E † , respectively. Date : June, 2020. After the actual Talbot’s Curve, shown in Figure 1 (bottom): the negative pedal curve of anellipse with respect to its center O . Figure 1.
Examples of the Negative-Pedal Curve (light blue) of an ellipse (black) with respect toa point M . These are the envelope of lines through P ( t ) on the ellipse, perpendicular to P ( t ) − M ( Q n is the tangent point). Three cases are shown, for M (i) interior (top left), (ii) on the boundary(top right), and (iii) at the center (bottom) of the ellipse. For case (ii) the area of the curve isinvariant for all M [4]. Case (iii) yields Talbot’s Curve [11] (in general it does not pass throughthe foci, but for the case shown, a/b = 2 , it does). Main Results.
In Section 2 we review a theorem by Jakob Steiner [9, 10] concern-ing the Curvature Centroid (Krümmungs-Schwerpunkt) of polygons; a corollaryis that A p , A c , and A θ are invariant for M along any circle concentric with E .Furthermore, we prove A µ also shares this property.In Section 3, we derive explicit expressions for A , A p , A c , A θ , A µ in terms of E ’s semi-axes ( a, b ) , M , µ and θ . We also show that (i) A p − A c = A , and (ii) A p − A θ = A sin θ .In Section 4 we prove that both A ∗ and A † are invariant for M on E .Appendices A, B contain propositions supporting results related to ellipse evo-lutes, and pedal-like curves, respectively. A table of all symbols used herein appearsin Appendix C. REA-INVARIANT PEDAL-LIKE CURVES DERIVED FROM THE ELLIPSE 3
Figure 2.
The Ellipse Pedal Curve E p (red) is the locus of the foot Q p of the perpendiculardropped from M onto the line through P ( t ) tangent to the ellipse. The Contrapedal Curve E c (green) is the locus of foot Q c of the perpendicular dropped from M onto the line through P ( t ) normal to the ellipse. It can also be regarded as the pedal curve to the ellipse evolute (brownastroid). Sturm and Steiner: Circular Area Isocurves
A 1823 Theorem by Sturm states that given a triangle, the area of the pedaltriangle with respect to a point M is constant for all M on a circle centered on thecircumcenter X [8, Thm. 7.28, page 221], Figure 5.In 1825 Steiner generalized it as follows: given a polygon with vertices P i , i =1 , . . .N , the area of its pedal polygon with respect to M is invariant for M on acircle centered on Steiner’s curvature centroid K , given by [10]:(1) K = P i sin(2 θ i ) P i P i sin(2 θ i ) where θ i are the internal angles, i = 1 , · · · , N . In the same publication Steiner alsoproves that the pedal polygon with respect to K has extremal area. Note for N = 3 , K = X as the latter has barycentrics of sin(2 θ i ) [7]. This is consistent with thefact that pedal polygons with respect to points on the circumcircle have constantarea (in fact they have zero area, their vertices lie on the Simson line [11, SimsonLine]).Steiner further generalized the above to the case of a closed plane curve C , byapproximating it with a polygon where N →∞ . Let the pedal curve C p of C withrespect to a point M be the locus of the foot of the perpendicular dropped from M onto a point P ( t ) on C for all t ; see Figure 6. With N →∞ , provided that thetotal curvature of C is non-zero (i.e., non-zero winding number), K becomes [10]: D. REZNIK, R. GARCIA, AND H. STACHEL
Figure 3.
The ellipse Contrapedal Curve E c for four distinct positions of M = [ x m , y m ] . Noticethe contrapedal is the pedal of the evolute (dashed black). We invite the reader to prove that (i)the curve’s two self intersections (ignoring the one at M ) always occur at [ x m , and [0 , y m ] and(ii) that it touches the evolute at either 2 (top row) or 4 (bottom row) locations. Figure 4.
Left:
The rotated pedal curve E θ (red) is the locus of Q θ , the foot of a perpendiculardropped from M onto a line through P ( t ) , along the θ -rotated tangent (light blue), in this case θ = 54 ◦ . A θ is invariant for M on a concentric circle (orange). Note E = E p and E π/ = E c , andin general, E θ is the pedal curve with respect to the θ -evolutoid (green), whose curvature centroid K is stationary at O . Right:
The interpolated pedal curve E µ (red) is the locus of Q µ , the affinecombination of pedal Q p and contrapedal Q c feet, here µ = 1 / . A µ is invariant provided M lieson a concentric circle (orange).REA-INVARIANT PEDAL-LIKE CURVES DERIVED FROM THE ELLIPSE 5 Figure 5.
Left : Sturm’s Theorem (1823) states that the area of the pedal triangle (red) of areference triangle (blue) is invariant for all points M lying a circles (dashed black) centered onthe circumcenter X . Right : in 1825, Steiner generalizes this to polygons: the area of the pedalpolygon (red) to an N-gon (blue) is constant for all M over a circle centered on K , the curvaturecentroid. C denotes the polygon’s center of area. Figure 6.
A generic closed curve C (blue) and its pedal curve (red), defined as the locus of thefoot M ′ of perpendiculars dropped from M onto point P ( t ) on C . The Steiner curvature centroid K is obtained by averaging the curvature κ ( t ) over all P ( t ) ; see equation (2). The signed area ofthe pedal polygon is constant for M over a circle centered on K . (2) K = R κ ( s ) P ( s ) .ds R κ ( s ) .ds where κ ( s ) is the curvature and s is arc length. Referring to Figure 6, we recall aresult by Jakob Steiner [10]: Theorem (Steiner, 1825) . The area of the pedal curve is constant over points M lying on circles centered on K . From symmetry:
Lemma 1.
For the ellipse and its evolute (an astroid), K = O . D. REZNIK, R. GARCIA, AND H. STACHEL
Figure 7.
Left:
The areas A p (resp. A c ) of the Pedal Curve E p (red) (resp. Contrapedal Curve E c , green) is invariant over all M on a circle (orange) concentric with the ellipse. Right:
Aniso-area concentric circle (orange) of radius larger than the minor axis of the ellipse.
Note: when expressed in line coordinates, the cusps of the evolute are regular.Specifically, cusps of the evolute are inflection points of its dual [3, 2].Referring to Figure 7:
Corollary 1.
The area A p of the pedal curve is invariant for M on a circle con-centric with E . As illustrated for an ellipse in Figure 3, in general, the contrapedal curve is thepedal curve with respect to the evolute [11, Contrapedal Curve] and:
Corollary 2.
The area A c of the contrapedal curve is invariant for M on a circleconcentric with E . Let the term θ -evolutoid denote the envelope of θ -rotated tangents to a curve;see Figure 8. Lemma 2.
The θ -evolutoid to an ellipse has K = O , for any θ . This stems from the fact that for all θ , the θ -evolutoid remains symmetric withrespect to the origin O . Corollary 3.
The area A θ of the rotated contrapedal curve is invariant for M ona circle concentric with E . This stems from the fact that the rotated pedal curve E θ is the pedal with respectto a θ -evolutoid and Lemma 2. Theorem 1.
The area A µ of the interpolated pedal curve is invariant for M on acircle concentric with E .Proof. Proposition 9 in Appendix B shows that for any closed curve with non-zerototal curvature (the denominator of Equation (2)), A µ is a fixed linear function of A p and A c . Since both A p and A c are constant for M on circles centered on K = O ,the result follows. (cid:3) REA-INVARIANT PEDAL-LIKE CURVES DERIVED FROM THE ELLIPSE 7
Figure 8.
The θ -evolutoid (purple, envelope of tangents rotated by θ ) for an a/b = 1 . ellipse, θ = 10 , . . . , degrees. The bottom-right figure ( θ = 90 ◦ ) is the ellipse evolute. Also shown arethe normal (black arrow) and rotated tangent (red arrow) for a point in the 1st quadrant. Sincethese curves are centrally symmetric, the Steiner curvature centroid K lies at the ellipse center. Explicit Areas
As before, let a point P ( t ) on E be parametrized as P ( t ) = ( a cos t, b sin t ) . Definethe signed area of a curve γ as:(3) A γ = 12 Z γ ( xdy − ydx ) . Referring to Figure 8, the θ -evolutoid is the envelope of lines passing through P ( t ) rotated with respect to the tangent vector P ′ ( t ) by θ . Its coordinates can bederived explicitly as x θ ( t ) = a cos θ cos t + c sin θ cos ta − sin t sin θ cos θ ( b cos t + a sin t ) by θ ( t ) = a sin θ cos θ cos t − c sin θ cos t ( b cos t cos θ − a sin θ sin t ) ab + sin t (cid:0) b cos θ − c sin θ (cid:1) b with c = a − b .Let θ = tan − (cid:0) ab c (cid:1) . Remark . The θ -evolutoid will have 4, 2, or 0 singularities if θ ∈ ( θ , π − θ ) , θ ∈ { θ , π − θ } , or θ / ∈ [ θ , π − θ ] , respectively. Moreover, the θ -evolutoid issingular at t = π and t = π . D. REZNIK, R. GARCIA, AND H. STACHEL
Proposition 1.
The signed area S θ of the θ -evolutoid is given by: S θ = π ab cos θ − c ab sin θ Proof.
Direct integration of Equation 3. (cid:3)
Let M = ( x , y ) . Proposition 2.
The areas A p and A c of E p and E c are given by: A p = π (cid:0) a + b + x + y (cid:1) (4) A c = π (cid:0) a + b − ab + x + y (cid:1) Proof.
Consider the ellipse parametrized by P ( t ) = ( a cos t, b sin t ) . Then it followsthat E p ( t ) = (cid:20) ( a x sin t − aby cos t sin t + ab cos t ) b cos t + a sin t , ( b y cos t − abx cos t sin t + a b sin tb cos t + a sin t (cid:21) E c ( t ) = (cid:20) b x cos t + cos t sin t ( aby + ac sin t ) b cos t + a sin t , a y sin t + cos t sin t ( abx − bc cos t ) b cos t + a sin t (cid:21) Compute the above areas with Equation 3. The integrand will be a ratio oftrigonometric polynomials. Evaluate the integrals by using classical residue theory[1]. Algebraic manipulation yields the claim. (cid:3)
Note: formulas in (4) and later are consistent with Steiner’s result that the area ofthe pedal curve of M is the sum of the area for M = K and a term proportionalto the square of | M K | [10, p. 47]. Corollary 4. A p − A c = A . Note the above holds holds for all convex curves, as proved in Proposition 7, Ap-pendix B.
Proposition 3.
The area A θ of the rotated pedal curve is given by A θ = π (cid:0) a + b − ab sin θ + x + y (cid:1) Proof.
Similar to Proposition 2. (cid:3)
Corollary 5. A p − A θ = A sin θ . Proposition 4.
The area A µ of E µ = (1 − µ ) E p + µ E c is given by A µ = (cid:0) a − ab + 2 b + 2 x + 2 y (cid:1) π µ − (cid:0) ( a − b ) + x + y (cid:1) π µ + 12 (cid:0) ( a − b ) + x + y (cid:1) π =(2 µ − − µ ) A p − µA c ] + µ (1 − µ ) A. Proof.
Similar to Proposition 2. (cid:3)
REA-INVARIANT PEDAL-LIKE CURVES DERIVED FROM THE ELLIPSE 9 Area Invariance of Hybrid and Pseudo Talbot Curves
As defined in Section 1, let (i) the Hybrid Pedal Curve E ∗ be the locus of theintersection Q ∗ of L ( t ) with the line from M to Q p , Figure 9, and (ii) the PseudoTalbot Curve E † be the Negative Pedal Curve of E ∗ , Figure 10. Here we prove theirarea invariance over all M on E . Theorem 2.
The area A ∗ of E ∗ is invariant for all M on E and given by: A ∗ = π (3 a + 2 a b + 3 b )2 ab = π (3 δ + 5 a b )2 ab , δ = p a − a b + b Proof.
Let P ( t ) = [ a cos t, b sin t ] be a point on E . By definition (Section 1 andFigure 9), E ∗ is defined as the intersection of lines M + uP ′ ( t ) ⊥ and P ( t ) + v ( M − P ( t )) ⊥ . let M = ( x , y ) and E ( t ) = [ x ∗ ( t ) , y ∗ ( t )] . Straightforward calculation leads to: x ∗ ( t ) = − b (cid:0) a + b + 4 y (cid:1) cos t + 4 abx cos 2 t − bc cos 3 t + 4 ax y sin t + 4 b y sin 2 t ay sin t + bx cos t − ab ) y ∗ ( t ) = − a (cid:0) a + 3 b + 4 x (cid:1) sin t + 4 a x sin 2 t − ac sin 3 t + 4 bx y cos t − aby cos 2 t ay sin t + bx cos t − ab ) where c = a − b . Integrating Equation 3 over E ∗ ( t ) yields the claim. (cid:3) Theorem 3.
The area A † of E † is invariant for all M on E and given by: A † = π (cid:0) a + 2 a b + 3 b (cid:1) (cid:0) a − ab − b (cid:1) (cid:0) a + 2 ab − b (cid:1) a b Proof.
Recall E † is the negative pedal curve of E ∗ (Section 1 and Figure 10). For M on the ellipse, the coordinates [ x † , y † ] of E † can be derived explicitly: x † ( u ) = − (cid:0) ( k x + 1) a − k x + 1) b a + k x b (cid:1) cos ub a − (cid:0) a − b (cid:1) sin t cos t sin ub a + cos t (cid:0) a + b (cid:1) (cid:0) − a sin t − b cos t + 2 b (cid:1) b ay † ( u ) = − (cid:0) a − b (cid:1) sin t cos t cos ua b − (cid:0) ( k y − a − k y b a + k y b (cid:1) sin ua b + sin t (cid:0) a + b (cid:1) (cid:0) ( a − b ) cos t + a (cid:1) a bk x =2 cos t − tk y =2 cos t − cos t Integrating Equation (3) for the above yields the claimed results. (cid:3) Conclusion
One open question is whether a common thread exists which links the SteinerHat [4], the Hybrid, and Pseudo-Talbot curves, since all of them are area-invariantover M on the ellipse. Furthermore, if a continuous family of curves exists withthis area-invariance property. Figure 9.
The Pedal E p (red) and Hybrid Pedal E ∗ (light brown), and Negative-Pedal E n (lightblue) Curves of the ellipse E (black) shown for two positions of M . E ∗ is the locus of the intersection Q ∗ of (i) the line through P ( t ) perpendicular to P ( t ) − M and (ii) line MQ p (note E p is the locusof Q p ). For both left and right pictures (indeed for all M on the ellipse), areas A n , A ∗ are constant,thought that of E p varies. Also shown are iso-curves of constant A ∗ inside the ellipse; these arehigh-order rational curves. E ∗ is unstable when M is exterior to E . Acknowledgments
We would like to thank Robert Ferréol and Mark Helman for their help duringthis work.The second author is fellow of CNPq and coordinator of Project PRONEX/CNPq/ FAPEG 2017 10 26 7000 508.
Appendix A. Evolutoids
Consider a plane convex curve C ( t ) = ( x ( t ) , y ( t )) defined by a support function h : x ( t ) = h ( t ) cos t − h ′ ( t ) sin t (5) y ( t ) = h ( t ) sin t + h ′ ( t ) cos t The family of lines passing through P ( t ) = ( x ( t ) , y ( t )) making a constant angle θ with ( x ′ ( t ) , y ′ ( t )) is given by: REA-INVARIANT PEDAL-LIKE CURVES DERIVED FROM THE ELLIPSE 11
Figure 10.
The pedal, negative pedal, hybrid pedal E ∗ , and pseudo-Talbot curve E † (negativepedal curve of E ∗ ) are shown red, light blue, brown, and olive green, respectively, for aspect ratios a/b of E of 1.5 (left) and 2.0 (right), respectively. Notice that for the former case E † has two cusps,and in the latter 4. Excluding the pedal curve, all other 3 are area-invariant over M on E , and areall tangent to E at the point where the normal goes thru M . L θ ( t ) : (sin (2 θ ) + sin (2 t )) x + (cos (2 θ ) − cos (2 t )) y − h ( t )(sin t − sin (2 θ + t )) + h ′ ( t )(cos t − cos (2 θ + t )) = 0 Let C θ ( t ) = ( x θ , y θ ) denote the envelope of L θ ( t ) . This will be given by: x θ ( t ) = 12 (cos ( t − θ ) + cos t ) h ( t ) − h ′ ( t ) sin t + 12 (cos ( t − θ ) − cos t ) h ′′ ( t ) y θ ( t ) = 12 (sin ( t − θ ) + sin t ) h ( t ) + h ′ ( t ) cos t + 12 (sin ( t − θ ) − sin t ) h ′′ ( t ) Note that C π/ is the evolute of C . Let h θ ( t ) = h ( t − θ ) cos θ + h ′ ( t − θ ) sin θ .Changing variables t = t − θ it follows that the envelope is given by x θ ( t − θ ) = h θ ( t ) cos t − h ′ θ ( t ) sin ty θ ( t − θ ) = h θ ( t ) sin t + h ′ θ ( t ) cos t Let S ( . ) denote the signed area of a curve. Then S ( C ) = 12 Z π ( h ( t ) − h ′ ( t ) ) dtS ( C π/ ) = 12 Z π ( h ′ ( t ) − h ′′ ( t ) ) dt Proposition 5. S ( C θ ) is given by S ( C θ ) = S ( C ) cos θ + S ( C π/ ) sin θ Proof.
The signed area of the evolute C π/ is negative in general, and zero if C is acircle. Integrating Equation 3 by parts and simplifying it yields the claim. (cid:3) Let L ( . ) denote the perimeter of a curve. Proposition 6.
For small θ , L ( C θ ) is given by: L ( C θ ) = L ( C ) cos θ Proof.
Let
T, N define the tangent and normal axis of the Frenet frame. From [5]we have that C θ ( s ) = C ( s ) + cos θ sin θk ( s ) T ( s ) + sin θk ( s ) N ( s ) . Differentiating the above and using Frenet equations T ′ = kN and N ′ = − kT ,it follows that C ′ θ ( s ) = k ( s ) cos θ − k ′ ( s ) sin θk ( s ) ( N ( s ) sin θ + T ( s ) cos θ ) Therefore, |C ′ θ ( s ) | = | cos θ − k ′ ( s ) k ( s ) sin θ | . Integration leads to the result stated. (cid:3)
Appendix B. Pedal and Contrapedal Areas
Let M = ( x , y ) be a fixed point. Referring to Equation 5, the pedal of M isgiven by(6) P M ( t ) = [ x sin t + ( h ( t ) − y sin t ) cos t, ( h ( t ) − x cos t ) sin t + y cos t ] . The contrapedal of M is given by(7) C M ( t ) = [ x cos t + y cos t sin t − h ′ ( t ) sin t, y sin t + x cos t sin t + h ′ ( t ) cos t ] . Below is a generalization of Corollary 4.
Proposition 7.
For all convex curves, the following holds: A ( P M ) − A ( C M ) = A ( C ) REA-INVARIANT PEDAL-LIKE CURVES DERIVED FROM THE ELLIPSE 13
Proof.
Obtain the signed areas for the above curves above via integration by partsof Equation 3. Algebraic manipulation yields the claim. In fact, A ( P M ) = π x + y ) − (cid:18)Z π h ( t ) cos tdt (cid:19) x − (cid:18)Z π h ( t ) sin tdt (cid:19) y + 12 Z π h ( t ) dtA ( C M ) = π x + y ) − (cid:18)Z π h ( t ) cos tdt (cid:19) x − (cid:18)Z π h ( t ) sin tdt (cid:19) y + 12 Z π h ′ ( t ) dtA ( C ) = 12 Z π ( h ( t ) − h ′ ( t ) ) dt (cid:3) Corollary 6.
The family of isocurves of A ( P M ) and A ( C M ) are circles centeredat K = (cid:18) π Z π h ( t ) cos t dt, π Z π h ( t ) sin t dt (cid:19) Proof.
Direct from the definition of the centroid K = π R π C ( t ) dt and expressionsof the areas A ( P M ) and A ( C M ) . (cid:3) Proposition 8.
Let P θM the pedal of M with respect to the curve C θ . For anyconvex curve C we have that: A ( P M ) − A ( P θM ) = sin θA ( C ) Proof.
Similar to the that of Proposition 7. (cid:3)
Referring to Figure 11 and generalizing Proposition 4:
Proposition 9.
For any smooth regular closed curve C with non-zero rotating index,the isocurves of A ( C µ ) are circles centered on K . In fact, A ( C µ ) = (2 µ − − µ ) A ( P M ) − µA ( C M )] + µ (1 − µ ) A ( C ) . Proof.
Consider a regular closed curve C ( s ) = ( x ( s ) , y ( s )) parametrized by arclength s and of length L . Let M = ( x , y ) . Write C ′ ( s ) = (cos θ ( s ) , sin θ ( s )) . Therefore, the curvature is k ( s ) = θ ′ ( s ) .Then, the pedal P M and contrapedal C M curves with respect ot M are given by P M =[( x − x ( s )) cos θ ( s ) + sin θ ( s ) cos θ ( s )( y − y ( s )) + x ( s ) , ( y ( s ) − y ) cos θ ( s ) + sin θ ( s ) cos θ ( s )( x − x ( s )) + y ] C M =[ x + ( x ( s ) − x ) cos θ ( s ) + ( y ( s ) − y ) cos θ ( s ) sin θ ( s ) ,y ( s ) + ( y − y ( s )) cos θ ( s ) + ( x ( s ) − x ) cos θ ( s ) sin θ ( s ) . ] Then, A ( P M ) = 14 Z L k ( s ) ds (cid:16) x + y (cid:17) − (cid:18)Z L k ( s ) x ( s ) ds (cid:19) x − (cid:18)Z L k ( s ) y ( s ) ds (cid:19) y + 12 Z L [cos θ ( s ) y ( s ) − sin θ ( s ) x ( s )] k ( s ) dsA ( C M ) = 14 Z L k ( s ) ds (cid:16) x + y (cid:17) − (cid:18)Z L k ( s ) x ( s ) ds (cid:19) x − (cid:18)Z L k ( s ) y ( s ) ds (cid:19) y + 12 Z L [cos θ ( s ) x ( s ) + sin θ ( s ) y ( s )] k ( s ) ds Therefore, A ( P M ) − A ( C M ) = − Z L (cid:16) x ( s ) y ( s ) sin 2 θ ( s ) + ( x ( s ) − y ( s ) ) cos 2 θ ( s ) (cid:17) k ( s ) dsA ( P M ) + A ( C M ) = 12 Z L k ( s ) ds (cid:16) x + y (cid:17) − (cid:18)Z L k ( s ) x ( s ) ds (cid:19) x − (cid:18)Z L k ( s ) y ( s ) ds (cid:19) y + 12 Z L ( x ( s ) + y ( s ) ) k ( s ) ds Let C µ = (1 − µ ) P M + µC M .Then, A ( C µ ) =(2 µ − (cid:20)(cid:18) Z L k ( s ) ds (cid:19) ( x + y ) − (cid:18) Z L k ( s ) x ( s ) ds (cid:19) x − (cid:18) Z L k ( s ) y ( s ) ds (cid:19) y (cid:21) + (2 µ − Z L h x ( s ) y ( s ) sin 2 θ ( s ) + cos 2 θ ( s )( x ( s ) − y ( s ) ) i k ( s ) ds + (2 µ − Z L ( x ( s ) + y ( s ) ) k ( s ) ds + Z L µ ( µ −
1) (cos θ ( s ) y ( s ) − x ( s ) sin θ ( s )) ds =(2 µ − − µ ) A ( P M ) − µA ( C M )] + µ (1 − µ ) A ( C ) . (cid:3) Remark.
When R L k ( s ) ds = 0 , or equivalently the rotating index of the curve iszero, the Steiner curvature centroid K is not defined. In this case the pedal andcontrapedal area isocurves will be either parallel lines or independent of M . Appendix C. Table of Symbols symbol meaning note E ellipse semi-axes a, b K r circle of radius r concentric with E M a point in the plane P ( t ) a point on E [ a cos t, b sin t ] L ( t ) line through P ( t ) along [ P ( t ) − M ] ⊥ Q p , Q c , Q θ pedal, contrapedal, rotated pedal feet Q µ linear interpolation of Q p , Q cp Q ∗ intersection of pedal line with L ( t ) E p pedal curve of E wrt M locus of Q p E n negative pedal curve of E wrt M envelope of L ( t ) E c contrapedal curve of E wrt M locus of Q c E θ rotated pedal curve of E wrt M locus of Q θ E µ interpolated pedal curve of E wrt M locus of Q µ E ∗ hybrid pedal curve of E wrt M locus of Q ∗ E † pseudo Talbot’s curve of E wrt M locus of Q † A area of E πabA p , A c , A θ , A µ areas of E p , E c , E θ , E µ invariant for M on a K r A n , A ∗ areas of E n , E ∗ invariant for M on E Table 1.
Symbols used.REA-INVARIANT PEDAL-LIKE CURVES DERIVED FROM THE ELLIPSE 15
Figure 11.
Top left : a generic concave curve C (black) is shown as well as its curvature centroid K and a circle about it where M lies. Also shown are the pedal (red) and contrapedal (green)curves with respect to M . Note their areas are invariant for M anywhere on a circle centered on K . Top right, bottom left, bottom right : the interpolated pedal curve (blue) for µ = 0 . , . , and . , respectively. Its area is also invariant for M on a circle centered on K . Notice thatat µ = 0 . (bottom left) the interpolated pedal is homothetic (scale of 1/2) to C and its shape(and area) is independent of the location of M . References [1] Ahlfors, L. V. (1979).
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