Cissoid of Diocles
Encyclopedia
In geometry
, the cissoid of Diocles is a cubic plane curve
notable for the property that it can be used to construct two mean proportional
s to a given ratio
. In particular, it can be used to double a cube
. It can be defined as the cissoid
of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the family of cissoids is named for this example and some authors refer to it simply as the cissoid. It has a single cusp
at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze
family of curves and in form it resembles a tractrix
.
The word "cissoid" comes from the Greek
κισσοείδες kissoeidēs "ivy
shaped" from κισσός kissos "ivy" and -οειδές -oeidēs "having the likeness of". The curve is named for Diocles
who studied it in the 2nd century BCE.
Let the radius of C be a. By translation and rotation, we may take O to be the origin and the center of the circle to be (a, 0), so A is (2a, 0). Then the polar equations of L and C are:
.
By construction, the distance from the origin to a point on the cissoid is equal the difference between the distances between the origin and the corresponding points on L and C. In other words, the polar equation of the cissoid is
.
Applying some trigonometric identities, this is equivalent to
.
Let
in the above equation. Then
are parametric equations for the cissoid.
Converting the polar form to Cartesian coordinates produces
To see this, let O be the origin and L the line x=2a as above. Let P be the point (2a, 2at), then Q is (0, 2at) and the equation of the line OP is y=tx. The line through Q perpendicular to OP is
.
To find the point of intersection R, set y=tx in this equation to get
which are the parametric equations given above.
This construction suggests the mechanism shown to the right as a way to generate the curve.
. Let J be a line and B a point not on J. Let BST be a right angle which moves so that ST equals the distance from B to J and T remains on J, while the other leg BS slides along B. Then the midpoint P of ST describes the curve.
To see this, let the distance between B and J be 2a. By translation and rotation, take B = (−a, 0) and J the line x=a. Let P = (x, y) and let ψ be the angle between SB and the x-axis; this is equal to the angle between ST and J. By construction, PT = a, so the distance from P to J is a sin ψ. In other words a-x = a sin ψ. Also, SP = a is the y coordinate of (x, y) if it is rotated by angle ψ, so
a = (x+a) sin ψ + y cos ψ. After simplification, this produces parametric equations
Change parameters by replacing ψ with its compliment to get
or, applying double angle formulas,
But this is polar equation
given above with θ=Ψ/2.
Note that, as with the double projection construction, this can be adapted to produce a mechanical device that generates the curve.
. This means that given lengths a and b, the curve can be used to find u and v so that
u/a=v/u=b/v. As a special case, this can be used to solve the Delian problem: how much must the length of a cube
be increased in order to double
its volume
? Specifically, if a is the side of a cube, and b=2a, then the volume of a cube of side u is
so u is the side of a cube with double the volume of the original cube. Note however that this solution does not fall within the rules of compass and straightedge construction since it relies on the existence of the cissoid.
Let a and b be given. It is required to find u so that u3=a2b, giving u and v=u2/a as the mean proportionals. Let the cissoid
be constructed as above, with O the origin, A the point (2a, 0), and J the line x=a, also as given above. Let C be the point of intersection of J with OA. From the given length b, mark B on J so that CB=b. Draw BA and let P = (x, y) be the point where it intersects the cissoid. Draw OP and let it intersect J at U. Then u=CU is the required length.
To see this, rewrite the equation of the curve as
and let N = (x, 0), so PN is the perpendicular to OA through P.
From the equation of the curve,
From this,
By similar triangles PN/ON=UA/OA and PM/MA=BO/OA. So the equation becomes
so
as required.
Diocles did not really solve the Delian problem. The reason is that the cissoid of Diocles cannot be constructed perfectly, at least not with compass and straightedge. To construct the cissoid of Diocles, one would construct a finite number of its individual points, then connect all these points to form a curve. The problem is that there is no well-defined way to connect the points. If they are connected by line segments, then the construction will be well-defined, but it will not be an exact cissoid of Diocles, but only an approximation. Likewise, if the dots are connected with circular arcs, the construction will be well-defined, but incorrect. Or one could simply draw a curve directly, trying to eyeball the shape of the curve, but the result would only be imprecise guesswork.
Once the finite set of points on the cissoid have been drawn, then line PC will probably not intersect one of these points exactly, but will pass between them, intersecting the cissoid of Diocles at some point whose exact location has not been constructed, but has only been approximated. An alternative is to keep adding constructed points to the cissoid which get closer and closer to the intersection with line PC, but the number of steps may very well be infinite, and the Greeks did not recognize approximations as limits of infinite steps (so they were very puzzled by Zeno's paradoxes
).
One could also construct a cissoid of Diocles by means of a mechanical tool specially designed for that purpose, but this violates the rule of only using compass and straightedge. This rule was established for reasons of logical — axiomatic — consistency. Allowing construction by new tools would be like adding new axiom
s, but axioms are supposed to be simple and self-evident, but such tools are not. So by the rules of classical, synthetic geometry
, Diocles did not solve the Delian problem, which actually can not be solved by such means.
On the other hand, if one accepts that cissoids of Diocles do exist
, then there must exist at least one example of such a cissoid. This cissoid could then be translated, rotated, and expanded or contracted in size (without changing its proportional
shape) at will to fit into any position. Then one would readily admit that such a cissoid can be used to correctly solve the Delian problem.
. Take two congruent parabola
s, set them vertex-to-vertex, and roll one along the other; the vertex of the rolling parabola will trace the cissoid.
Define a pair of parabolas whose equations are
These parabolas face each other symmetrically across the x-axis, and are the ones shown in Figure 1. Then pick a point (a, a2) from the top parabola. The slope
of the tangent to the top parabola at this point is
The distance measured along the top parabola from point (0,0) to point (a, a2) is equivalent to the distance measured along the bottom parabola from point (0,0) to point (a, −a2). Thus, when the top parabola rolls, its point which was originally at (a, a2) will eventually come into tangential contact with the point (a, −a2) of the bottom parabola. This is true for all values of a.
When the two such points are in contact, their tangent
s coincide and have the same slope, which is −2 a. Let θ be the angle which the slope 2 a makes with the x-axis, so that
.
Then the top parabola must rotate by an angle of 2 θ in order for its point originally at (a, a2) to come into contact with the bottom parabola's point at (a, −a2).
This rotation can be performed by means of the rotation matrix R(−2 θ), viz.
Let the pivoting point of the rotation be point (a, a2). This pivoting vector must be subtracted from all the points on the top parabola before all these points are operated upon by the rotation matrix. Then, after the rotation is performed, this pivoting point will be at the origin, and the slope of the rotated top parabola at the origin will be −2 a, which corresponds to the slope of the point (a, −a2) of the bottom parabola. These two points must come into contact, so the rotated top parabola will be moved by vector (a, −a2) to its new position in contact with the bottom parabola. Thus the rolled top parabola will be described by
where a may be called a "rolling parameter" and x is the pre-rotated abscissa
.
Applying the trigonometric identities
yields a locus of points of the form
which, through application of equation (1), becomes
Then, applying the identities
produces the set of points of the form
The vertex
of the rolled top parabola is specified by letting x = 0, which results in the set of points which depend only on the rolling parameter a and whose coordinates are
This curve of points (X(a), Y(a)) is the roulette, and now it only remains to be shown that it is a cissoid of Diocles. The roulette is a cissoid whose equation is
.
To demonstrate this, plug in the values of X and Y given in equations (2) and (3) into equation (4) and see if it leads to a tautology
:
which is true for all values of a, the rolling parameter.
of a parabola with respect to its vertex is a cissoid of Diocles.
Proof: Any parabola can be rotated and translated so that it will end up being described by the equation
whose slope at point (x, y) is given by the derivative
Then the set of points which form the line tangent to the parabola at point (b, a b2) is
and the set of points which form the line which is perpendicular to L1(b) and which passes through the origin
is
Notice that L2(b)′s slope is −1/(2ab) which is perpendicular to L1(b)′s slope 2ab.
The intersection L1(b) ∩ L2(b) can be found by setting up the following system of equations
Now solve for u:
so that the abscissa
v is
and the ordinate is
Thus, the point of intersection is
and this point belongs to the pedal curve of the parabola.
Let
Then the question is: is there a constant k which does not depend on b (but only on a) such that
?
To find out, substitute the values for X and Y:
Multiply the numerator and denominator on the right side by (4 a2 b2 + 1)3,
Cross-multiply,
Divide by (4 a2 b2 + 1)2,
Distribute
,
Divide both sides by a3 b6,
Subtract 4 a2 b2 from both sides, then factor out 8 a k on the left side,
Cancel out the common factor on both sides, then solve for k,
k does not depend on b, so, for a given parabola (with fixed value of a), k is a constant, and the pedal curve is a cissoid of Diocles.
of a parabola with the center of inversion at the vertex. To see this, take the parabola to be x=y2. In polar coordinates this becomes
,
and the inverse curve then has equation
which is a special case of the equation defining the cissoid of Diocles on polar coordinates.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the cissoid of Diocles is a cubic plane curve
Cubic plane curve
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equationapplied to homogeneous coordinates x:y:z for the projective plane; or the inhomogeneous version for the affine space determined by setting z = 1 in such an equation...
notable for the property that it can be used to construct two mean proportional
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s to a given ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
. In particular, it can be used to double a cube
Doubling the cube
Doubling the cube is one of the three most famous geometric problems unsolvable by compass and straightedge construction...
. It can be defined as the cissoid
Cissoid
In geometry, a cissoid is a curve generated from two given curves C1, C2 and a point O . Let L be a variable line passing through O and intersecting C1 at P1 and C2 at P2. Let P be the point on L so that OP = P1P2...
of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the family of cissoids is named for this example and some authors refer to it simply as the cissoid. It has a single cusp
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze
Conchoid of de Sluze
The conchoid of de Sluze is a family of plane curves studied in 1662 by René François Walter, baron de Sluze.The curves are defined by the polar equationr=\sec\theta+a\cos\theta \,....
family of curves and in form it resembles a tractrix
Tractrix
Tractrix is the curve along which a small object moves, under the influence of friction, when pulled on a horizontal plane by a piece of thread and a puller that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of...
.
The word "cissoid" comes from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
κισσοείδες kissoeidēs "ivy
Ivy
Ivy, plural ivies is a genus of 12–15 species of evergreen climbing or ground-creeping woody plants in the family Araliaceae, native to western, central and southern Europe, Macaronesia, northwestern Africa and across central-southern Asia east to Japan and Taiwan.-Description:On level ground they...
shaped" from κισσός kissos "ivy" and -οειδές -oeidēs "having the likeness of". The curve is named for Diocles
Diocles (mathematician)
Diocles was a Greek mathematician and geometer.- Life and work :Although little is known about the life of Diocles, it is known that he was a contemporary of Apollonius and that he flourished sometime around the end of the 3rd century BC and the beginning of the 2nd century BC.Diocles is thought...
who studied it in the 2nd century BCE.
Construction and equations
Let C be a circle with diameter OA and let L be the line tangent to the circle at A. Draw a variable ray R through O to intersect C at M1 and L at M2. Let M be the point on R so that OM = M1M2. The cissoid is then defined to be the locus of points M.Let the radius of C be a. By translation and rotation, we may take O to be the origin and the center of the circle to be (a, 0), so A is (2a, 0). Then the polar equations of L and C are:
.By construction, the distance from the origin to a point on the cissoid is equal the difference between the distances between the origin and the corresponding points on L and C. In other words, the polar equation of the cissoid is
.Applying some trigonometric identities, this is equivalent to
.Let
in the above equation. Thenare parametric equations for the cissoid.
Converting the polar form to Cartesian coordinates produces
Construction by double projection
Another construction proceeds as follows: Let a line L and a point O not on L be given, and let K be the line through O parallel to L. Let P be a variable point on L. Let Q be the projection of P onto K, on other words Q is the intersection of K with the line through P perpendicular to K. Similarly let R be the projection of Q onto OP. Then the cissoid is the locus of points R.To see this, let O be the origin and L the line x=2a as above. Let P be the point (2a, 2at), then Q is (0, 2at) and the equation of the line OP is y=tx. The line through Q perpendicular to OP is
.To find the point of intersection R, set y=tx in this equation to get
which are the parametric equations given above.
This construction suggests the mechanism shown to the right as a way to generate the curve.
Newton's construction
The following construction was given by Isaac NewtonIsaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
. Let J be a line and B a point not on J. Let BST be a right angle which moves so that ST equals the distance from B to J and T remains on J, while the other leg BS slides along B. Then the midpoint P of ST describes the curve.
To see this, let the distance between B and J be 2a. By translation and rotation, take B = (−a, 0) and J the line x=a. Let P = (x, y) and let ψ be the angle between SB and the x-axis; this is equal to the angle between ST and J. By construction, PT = a, so the distance from P to J is a sin ψ. In other words a-x = a sin ψ. Also, SP = a is the y coordinate of (x, y) if it is rotated by angle ψ, so
a = (x+a) sin ψ + y cos ψ. After simplification, this produces parametric equations
Change parameters by replacing ψ with its compliment to get
or, applying double angle formulas,
But this is polar equation
given above with θ=Ψ/2.
Note that, as with the double projection construction, this can be adapted to produce a mechanical device that generates the curve.
Delian problem
The Greek geometer Diocles used the cissoid to obtain two mean proportionals to a given ratioRatio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
. This means that given lengths a and b, the curve can be used to find u and v so that
u/a=v/u=b/v. As a special case, this can be used to solve the Delian problem: how much must the length of a cube
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
be increased in order to double
Doubling the cube
Doubling the cube is one of the three most famous geometric problems unsolvable by compass and straightedge construction...
its volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
? Specifically, if a is the side of a cube, and b=2a, then the volume of a cube of side u is
so u is the side of a cube with double the volume of the original cube. Note however that this solution does not fall within the rules of compass and straightedge construction since it relies on the existence of the cissoid.
Let a and b be given. It is required to find u so that u3=a2b, giving u and v=u2/a as the mean proportionals. Let the cissoid
be constructed as above, with O the origin, A the point (2a, 0), and J the line x=a, also as given above. Let C be the point of intersection of J with OA. From the given length b, mark B on J so that CB=b. Draw BA and let P = (x, y) be the point where it intersects the cissoid. Draw OP and let it intersect J at U. Then u=CU is the required length.
To see this, rewrite the equation of the curve as
and let N = (x, 0), so PN is the perpendicular to OA through P.
From the equation of the curve,
From this,
By similar triangles PN/ON=UA/OA and PM/MA=BO/OA. So the equation becomes
so
as required.
Diocles did not really solve the Delian problem. The reason is that the cissoid of Diocles cannot be constructed perfectly, at least not with compass and straightedge. To construct the cissoid of Diocles, one would construct a finite number of its individual points, then connect all these points to form a curve. The problem is that there is no well-defined way to connect the points. If they are connected by line segments, then the construction will be well-defined, but it will not be an exact cissoid of Diocles, but only an approximation. Likewise, if the dots are connected with circular arcs, the construction will be well-defined, but incorrect. Or one could simply draw a curve directly, trying to eyeball the shape of the curve, but the result would only be imprecise guesswork.
Once the finite set of points on the cissoid have been drawn, then line PC will probably not intersect one of these points exactly, but will pass between them, intersecting the cissoid of Diocles at some point whose exact location has not been constructed, but has only been approximated. An alternative is to keep adding constructed points to the cissoid which get closer and closer to the intersection with line PC, but the number of steps may very well be infinite, and the Greeks did not recognize approximations as limits of infinite steps (so they were very puzzled by Zeno's paradoxes
Zeno's paradoxes
Zeno's paradoxes are a set of problems generally thought to have been devised by Greek philosopher Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is...
).
One could also construct a cissoid of Diocles by means of a mechanical tool specially designed for that purpose, but this violates the rule of only using compass and straightedge. This rule was established for reasons of logical — axiomatic — consistency. Allowing construction by new tools would be like adding new axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s, but axioms are supposed to be simple and self-evident, but such tools are not. So by the rules of classical, synthetic geometry
Synthetic geometry
Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...
, Diocles did not solve the Delian problem, which actually can not be solved by such means.
On the other hand, if one accepts that cissoids of Diocles do exist
Existence
In common usage, existence is the world we are aware of through our senses, and that persists independently without them. In academic philosophy the word has a more specialized meaning, being contrasted with essence, which specifies different forms of existence as well as different identity...
, then there must exist at least one example of such a cissoid. This cissoid could then be translated, rotated, and expanded or contracted in size (without changing its proportional
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...
shape) at will to fit into any position. Then one would readily admit that such a cissoid can be used to correctly solve the Delian problem.
Roulette
This curve is also a rouletteRoulette (curve)
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....
. Take two congruent parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
s, set them vertex-to-vertex, and roll one along the other; the vertex of the rolling parabola will trace the cissoid.
- Figure 1. A pair of parabolas face each other symmetrically: one on top and one on the bottom. Then the top parabola is rolled without slipping along the bottom one, and its successive positions are shown in the animation. Then the path traced by the vertex of the top parabola as it rolls is a roulette shown in red, which happens to be a cissoid of Diocles.
Define a pair of parabolas whose equations are
These parabolas face each other symmetrically across the x-axis, and are the ones shown in Figure 1. Then pick a point (a, a2) from the top parabola. The slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
of the tangent to the top parabola at this point is
The distance measured along the top parabola from point (0,0) to point (a, a2) is equivalent to the distance measured along the bottom parabola from point (0,0) to point (a, −a2). Thus, when the top parabola rolls, its point which was originally at (a, a2) will eventually come into tangential contact with the point (a, −a2) of the bottom parabola. This is true for all values of a.
When the two such points are in contact, their tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
s coincide and have the same slope, which is −2 a. Let θ be the angle which the slope 2 a makes with the x-axis, so that
.Then the top parabola must rotate by an angle of 2 θ in order for its point originally at (a, a2) to come into contact with the bottom parabola's point at (a, −a2).
This rotation can be performed by means of the rotation matrix R(−2 θ), viz.
Let the pivoting point of the rotation be point (a, a2). This pivoting vector must be subtracted from all the points on the top parabola before all these points are operated upon by the rotation matrix. Then, after the rotation is performed, this pivoting point will be at the origin, and the slope of the rotated top parabola at the origin will be −2 a, which corresponds to the slope of the point (a, −a2) of the bottom parabola. These two points must come into contact, so the rotated top parabola will be moved by vector (a, −a2) to its new position in contact with the bottom parabola. Thus the rolled top parabola will be described by
where a may be called a "rolling parameter" and x is the pre-rotated abscissa
Abscissa
In mathematics, abscissa refers to that element of an ordered pair which is plotted on the horizontal axis of a two-dimensional Cartesian coordinate system, as opposed to the ordinate...
.
Applying the trigonometric identities
yields a locus of points of the form
which, through application of equation (1), becomes
Then, applying the identities
produces the set of points of the form
The vertex
Vertex (curve)
In the geometry of curves, a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature. Other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant...
of the rolled top parabola is specified by letting x = 0, which results in the set of points which depend only on the rolling parameter a and whose coordinates are
This curve of points (X(a), Y(a)) is the roulette, and now it only remains to be shown that it is a cissoid of Diocles. The roulette is a cissoid whose equation is
.To demonstrate this, plug in the values of X and Y given in equations (2) and (3) into equation (4) and see if it leads to a tautology
Tautology (logic)
In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
:
which is true for all values of a, the rolling parameter.
The cissoid of Diocles as a pedal curve
THEOREM: The pedal curvePedal curve
In the differential geometry of curves, a pedal curve is a curve derived by construction from a given curve ....
of a parabola with respect to its vertex is a cissoid of Diocles.
Proof: Any parabola can be rotated and translated so that it will end up being described by the equation
whose slope at point (x, y) is given by the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
Then the set of points which form the line tangent to the parabola at point (b, a b2) is
and the set of points which form the line which is perpendicular to L1(b) and which passes through the origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
is
Notice that L2(b)′s slope is −1/(2ab) which is perpendicular to L1(b)′s slope 2ab.
The intersection L1(b) ∩ L2(b) can be found by setting up the following system of equations
Now solve for u:
so that the abscissa
Abscissa
In mathematics, abscissa refers to that element of an ordered pair which is plotted on the horizontal axis of a two-dimensional Cartesian coordinate system, as opposed to the ordinate...
v is
and the ordinate is
Thus, the point of intersection is
and this point belongs to the pedal curve of the parabola.
Let
Then the question is: is there a constant k which does not depend on b (but only on a) such that
?To find out, substitute the values for X and Y:
Multiply the numerator and denominator on the right side by (4 a2 b2 + 1)3,
Cross-multiply,
Divide by (4 a2 b2 + 1)2,
Distribute
Distributivity
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
,
Divide both sides by a3 b6,
Subtract 4 a2 b2 from both sides, then factor out 8 a k on the left side,
Cancel out the common factor on both sides, then solve for k,
k does not depend on b, so, for a given parabola (with fixed value of a), k is a constant, and the pedal curve is a cissoid of Diocles.
Inversion
The cissoid of Diocles also be defined as the inverse curveInverse curve
In geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle with center O and radius k the inverse of a point Q is the point P for which P lies on the ray OQ and OP·PQ = k2...
of a parabola with the center of inversion at the vertex. To see this, take the parabola to be x=y2. In polar coordinates this becomes
,and the inverse curve then has equation
which is a special case of the equation defining the cissoid of Diocles on polar coordinates.