Plane curve constructed from a given curve and fixed point
In Euclidean geometry , for a plane curve C and a given fixed point O , the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point. The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal pc (the contrapedal coordinate ) even though it is not an independent quantity and it relates to (r , p ) as p c := r 2 − p 2 . {\textstyle p_{c}:={\sqrt {r^{2}-p^{2}}}.}
Some curves have particularly simple pedal equations and knowing the pedal equation of a curve may simplify the calculation of certain of its properties such as curvature . These coordinates are also well suited for solving certain type of force problems in classical mechanics and celestial mechanics .
Cartesian coordinates [ edit ] For C given in rectangular coordinates by f (x , y ) = 0, and with O taken to be the origin, the pedal coordinates of the point (x , y ) are given by:[ 1]
r = x 2 + y 2 {\displaystyle r={\sqrt {x^{2}+y^{2}}}} p = x ∂ f ∂ x + y ∂ f ∂ y ( ∂ f ∂ x ) 2 + ( ∂ f ∂ y ) 2 . {\displaystyle p={\frac {x{\frac {\partial f}{\partial x}}+y{\frac {\partial f}{\partial y}}}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}}}}.} The pedal equation can be found by eliminating x and y from these equations and the equation of the curve.
The expression for p may be simplified if the equation of the curve is written in homogeneous coordinates by introducing a variable z , so that the equation of the curve is g (x , y , z ) = 0. The value of p is then given by[ 2]
p = ∂ g ∂ z ( ∂ g ∂ x ) 2 + ( ∂ g ∂ y ) 2 {\displaystyle p={\frac {\frac {\partial g}{\partial z}}{\sqrt {\left({\frac {\partial g}{\partial x}}\right)^{2}+\left({\frac {\partial g}{\partial y}}\right)^{2}}}}} where the result is evaluated at z =1
For C given in polar coordinates by r = f (θ), then
p = r sin ϕ {\displaystyle p=r\sin \phi } where ϕ {\displaystyle \phi } is the polar tangential angle given by
r = d r d θ tan ϕ . {\displaystyle r={\frac {dr}{d\theta }}\tan \phi .} The pedal equation can be found by eliminating θ from these equations.[ 3]
Alternatively, from the above we can find that
| d r d θ | = r p c p , {\displaystyle \left|{\frac {dr}{d\theta }}\right|={\frac {rp_{c}}{p}},} where p c := r 2 − p 2 {\displaystyle p_{c}:={\sqrt {r^{2}-p^{2}}}} is the "contrapedal" coordinate, i.e. distance to the normal. This implies that if a curve satisfies an autonomous differential equation in polar coordinates of the form:
f ( r , | d r d θ | ) = 0 , {\displaystyle f\left(r,\left|{\frac {dr}{d\theta }}\right|\right)=0,} its pedal equation becomes
f ( r , r p c p ) = 0. {\displaystyle f\left(r,{\frac {rp_{c}}{p}}\right)=0.} As an example take the logarithmic spiral with the spiral angle α:
r = a e cos α sin α θ . {\displaystyle r=ae^{{\frac {\cos \alpha }{\sin \alpha }}\theta }.} Differentiating with respect to θ {\displaystyle \theta } we obtain
d r d θ = cos α sin α a e cos α sin α θ = cos α sin α r , {\displaystyle {\frac {dr}{d\theta }}={\frac {\cos \alpha }{\sin \alpha }}ae^{{\frac {\cos \alpha }{\sin \alpha }}\theta }={\frac {\cos \alpha }{\sin \alpha }}r,} hence
| d r d θ | = | cos α sin α | r , {\displaystyle \left|{\frac {dr}{d\theta }}\right|=\left|{\frac {\cos \alpha }{\sin \alpha }}\right|r,} and thus in pedal coordinates we get
r p p c = | cos α sin α | r , ⇒ | sin α | p c = | cos α | p , {\displaystyle {\frac {r}{p}}p_{c}=\left|{\frac {\cos \alpha }{\sin \alpha }}\right|r,\qquad \Rightarrow \qquad |\sin \alpha |p_{c}=|\cos \alpha |p,} or using the fact that p c 2 = r 2 − p 2 {\displaystyle p_{c}^{2}=r^{2}-p^{2}} we obtain
p = | sin α | r . {\displaystyle p=|\sin \alpha |r.} This approach can be generalized to include autonomous differential equations of any order as follows:[ 4] A curve C which a solution of an n -th order autonomous differential equation ( n ≥ 1 {\displaystyle n\geq 1} ) in polar coordinates
f ( r , | r θ ′ | , r θ ″ , | r θ ‴ | … , r θ ( 2 j ) , | r θ ( 2 j + 1 ) | , … , r θ ( n ) ) = 0 , {\displaystyle f\left(r,|r'_{\theta }|,r''_{\theta },|r'''_{\theta }|\dots ,r_{\theta }^{(2j)},|r_{\theta }^{(2j+1)}|,\dots ,r_{\theta }^{(n)}\right)=0,} is the pedal curve of a curve given in pedal coordinates by
f ( p , p c , p c p c ′ , p c ( p c p c ′ ) ′ , … , ( p c ∂ p ) n p ) = 0 , {\displaystyle f(p,p_{c},p_{c}p_{c}',p_{c}(p_{c}p_{c}')',\dots ,(p_{c}\partial _{p})^{n}p)=0,} where the differentiation is done with respect to p {\displaystyle p} .
Solutions to some force problems of classical mechanics can be surprisingly easily obtained in pedal coordinates.
Consider a dynamical system:
x ¨ = F ′ ( | x | 2 ) x + 2 G ′ ( | x | 2 ) x ˙ ⊥ , {\displaystyle {\ddot {x}}=F^{\prime }(|x|^{2})x+2G^{\prime }(|x|^{2}){\dot {x}}^{\perp },} describing an evolution of a test particle (with position x {\displaystyle x} and velocity x ˙ {\displaystyle {\dot {x}}} ) in the plane in the presence of central F {\displaystyle F} and Lorentz like G {\displaystyle G} potential. The quantities:
L = x ⋅ x ˙ ⊥ + G ( | x | 2 ) , c = | x ˙ | 2 − F ( | x | 2 ) , {\displaystyle L=x\cdot {\dot {x}}^{\perp }+G(|x|^{2}),\qquad c=|{\dot {x}}|^{2}-F(|x|^{2}),} are conserved in this system.
Then the curve traced by x {\displaystyle x} is given in pedal coordinates by
( L − G ( r 2 ) ) 2 p 2 = F ( r 2 ) + c , {\displaystyle {\frac {\left(L-G(r^{2})\right)^{2}}{p^{2}}}=F(r^{2})+c,} with the pedal point at the origin. This fact was discovered by P. Blaschke in 2017.[ 5]
As an example consider the so-called Kepler problem , i.e. central force problem, where the force varies inversely as a square of the distance:
x ¨ = − M | x | 3 x , {\displaystyle {\ddot {x}}=-{\frac {M}{|x|^{3}}}x,} we can arrive at the solution immediately in pedal coordinates
L 2 2 p 2 = M r + c , {\displaystyle {\frac {L^{2}}{2p^{2}}}={\frac {M}{r}}+c,} , where L {\displaystyle L} corresponds to the particle's angular momentum and c {\displaystyle c} to its energy. Thus we have obtained the equation of a conic section in pedal coordinates.
Inversely, for a given curve C , we can easily deduce what forces do we have to impose on a test particle to move along it.
Pedal equations for specific curves [ edit ] For a sinusoidal spiral written in the form
r n = a n sin ( n θ ) {\displaystyle r^{n}=a^{n}\sin(n\theta )} the polar tangential angle is
ψ = n θ {\displaystyle \psi =n\theta } which produces the pedal equation
p a n = r n + 1 . {\displaystyle pa^{n}=r^{n+1}.} The pedal equation for a number of familiar curves can be obtained setting n to specific values:[ 6]
n Curve Pedal point Pedal eq. All Circle with radius a Center p a n = r n + 1 {\displaystyle pa^{n}=r^{n+1}} 1 Circle with diameter a Point on circumference pa = r 2 −1 Line Point distance a from line p = a 1 ⁄2 Cardioid Cusp p 2 a = r 3 −1 ⁄2 Parabola Focus p 2 = ar 2 Lemniscate of Bernoulli Center pa 2 = r 3 −2 Rectangular hyperbola Center rp = a 2
A spiral shaped curve of the form
r = c θ α , {\displaystyle r=c\theta ^{\alpha },} satisfies the equation
d r d θ = α r α − 1 α , {\displaystyle {\frac {dr}{d\theta }}=\alpha r^{\frac {\alpha -1}{\alpha }},} and thus can be easily converted into pedal coordinates as
1 p 2 = α 2 c 2 α r 2 + 2 α + 1 r 2 . {\displaystyle {\frac {1}{p^{2}}}={\frac {\alpha ^{2}c^{\frac {2}{\alpha }}}{r^{2+{\frac {2}{\alpha }}}}}+{\frac {1}{r^{2}}}.} Special cases include:
α {\displaystyle \alpha } Curve Pedal point Pedal eq. 1 Spiral of Archimedes Origin 1 p 2 = 1 r 2 + c 2 r 4 {\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {c^{2}}{r^{4}}}} −1 Hyperbolic spiral Origin 1 p 2 = 1 r 2 + 1 c 2 {\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {1}{c^{2}}}} 1 ⁄2 Fermat's spiral Origin 1 p 2 = 1 r 2 + c 4 4 r 6 {\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {c^{4}}{4r^{6}}}} −1 ⁄2 Lituus Origin 1 p 2 = 1 r 2 + r 2 4 c 4 {\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {r^{2}}{4c^{4}}}}
Epi- and hypocycloids [ edit ] For an epi- or hypocycloid given by parametric equations
x ( θ ) = ( a + b ) cos θ − b cos ( a + b b θ ) {\displaystyle x(\theta )=(a+b)\cos \theta -b\cos \left({\frac {a+b}{b}}\theta \right)} y ( θ ) = ( a + b ) sin θ − b sin ( a + b b θ ) , {\displaystyle y(\theta )=(a+b)\sin \theta -b\sin \left({\frac {a+b}{b}}\theta \right),} the pedal equation with respect to the origin is[ 7]
r 2 = a 2 + 4 ( a + b ) b ( a + 2 b ) 2 p 2 {\displaystyle r^{2}=a^{2}+{\frac {4(a+b)b}{(a+2b)^{2}}}p^{2}} or[ 8]
p 2 = A ( r 2 − a 2 ) {\displaystyle p^{2}=A(r^{2}-a^{2})} with
A = ( a + 2 b ) 2 4 ( a + b ) b . {\displaystyle A={\frac {(a+2b)^{2}}{4(a+b)b}}.} Special cases obtained by setting b =a ⁄n for specific values of n include:
n Curve Pedal eq. 1, −1 ⁄2 Cardioid p 2 = 9 8 ( r 2 − a 2 ) {\displaystyle p^{2}={\frac {9}{8}}(r^{2}-a^{2})} 2, −2 ⁄3 Nephroid p 2 = 4 3 ( r 2 − a 2 ) {\displaystyle p^{2}={\frac {4}{3}}(r^{2}-a^{2})} −3, −3 ⁄2 Deltoid p 2 = − 1 8 ( r 2 − a 2 ) {\displaystyle p^{2}=-{\frac {1}{8}}(r^{2}-a^{2})} −4, −4 ⁄3 Astroid p 2 = − 1 3 ( r 2 − a 2 ) {\displaystyle p^{2}=-{\frac {1}{3}}(r^{2}-a^{2})}
Other pedal equations are:,[ 9]
Curve Equation Pedal point Pedal eq. Line a x + b y + c = 0 {\displaystyle ax+by+c=0} Origin p = | c | a 2 + b 2 {\displaystyle p={\frac {|c|}{\sqrt {a^{2}+b^{2}}}}} Point ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} Origin r = x 0 2 + y 0 2 {\displaystyle r={\sqrt {x_{0}^{2}+y_{0}^{2}}}} Circle | x − a | = R {\displaystyle |x-a|=R} Origin 2 p R = r 2 + R 2 − | a | 2 {\displaystyle 2pR=r^{2}+R^{2}-|a|^{2}} Involute of a circle r = a cos α , θ = tan α − α {\displaystyle r={\frac {a}{\cos \alpha }},\ \theta =\tan \alpha -\alpha } Origin p c = | a | {\displaystyle p_{c}=|a|} Ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1} Center a 2 b 2 p 2 + r 2 = a 2 + b 2 {\displaystyle {\frac {a^{2}b^{2}}{p^{2}}}+r^{2}=a^{2}+b^{2}} Hyperbola x 2 a 2 − y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1} Center − a 2 b 2 p 2 + r 2 = a 2 − b 2 {\displaystyle -{\frac {a^{2}b^{2}}{p^{2}}}+r^{2}=a^{2}-b^{2}} Ellipse x 2 a 2 + y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1} Focus b 2 p 2 = 2 a r − 1 {\displaystyle {\frac {b^{2}}{p^{2}}}={\frac {2a}{r}}-1} Hyperbola x 2 a 2 − y 2 b 2 = 1 {\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1} Focus b 2 p 2 = 2 a r + 1 {\displaystyle {\frac {b^{2}}{p^{2}}}={\frac {2a}{r}}+1} Logarithmic spiral r = a e θ cot α {\displaystyle r=ae^{\theta \cot \alpha }} Pole p = r sin α {\displaystyle p=r\sin \alpha } Cartesian oval | x | + α | x − a | = C , {\displaystyle |x|+\alpha |x-a|=C,} Focus ( b − ( 1 − α 2 ) r 2 ) 2 4 p 2 = C b r + ( 1 − α 2 ) C r − ( ( 1 − α 2 ) C 2 + b ) , b := C 2 − α 2 | a | 2 {\displaystyle {\frac {(b-(1-\alpha ^{2})r^{2})^{2}}{4p^{2}}}={\frac {Cb}{r}}+(1-\alpha ^{2})Cr-((1-\alpha ^{2})C^{2}+b),\ b:=C^{2}-\alpha ^{2}|a|^{2}} Cassini oval | x | | x − a | = C , {\displaystyle |x||x-a|=C,} Focus ( 3 C 2 + r 4 − | a | 2 r 2 ) 2 p 2 = 4 C 2 ( 2 C 2 r 2 + 2 r 2 − | a | 2 ) . {\displaystyle {\frac {(3C^{2}+r^{4}-|a|^{2}r^{2})^{2}}{p^{2}}}=4C^{2}\left({\frac {2C^{2}}{r^{2}}}+2r^{2}-|a|^{2}\right).} Cassini oval | x − a | | x + a | = C , {\displaystyle |x-a||x+a|=C,} Center 2 R p r = r 4 + R 2 − | a | 2 . {\displaystyle 2Rpr=r^{4}+R^{2}-|a|^{2}.}
^ Yates §1 ^ Edwards p. 161 ^ Yates p. 166, Edwards p. 162 ^ Blaschke Proposition 1 ^ Blaschke Theorem 2 ^ Yates p. 168, Edwards p. 162 ^ Edwards p. 163 ^ Yates p. 163 ^ Yates p. 169, Edwards p. 163, Blaschke sec. 2.1 R.C. Yates (1952). "Pedal Equations". A Handbook on Curves and Their Properties . Ann Arbor, MI: J. W. Edwards. pp. 166 ff. J. Edwards (1892). Differential Calculus . London: MacMillan and Co. pp. 161 ff.