Limaçon

The limaçon is a polar curve of the form ,

also called the limaçon of Pascal.The word "limaçon" comes from the Latin limax, meaning "snail."

If , the limaçon is convex. If

, the limaçon is dimpled. If

, the limaçon degenerates to a

cardioid. If , the limaçon has an

inner loop. If , it is a trisectrix (but not the Maclaurin trisectrix).

For , the inner loop has area

where . Similarly the area enclosed by the outer envelope is

Thus, the area between the loops is

In the special case of , these simplify to

Archimedes' Spiral

Archimedes' spiral is an Archimedean spiral with polar equation

This spiral was studied by Conon, and later by Archimedes in On Spirals about 225 BC. Archimedes was able to work out the lengths of various tangents to the spiral.

The curvature of Archimedes' spiral is

and the arc length is

Hyperbolic Spiral

An Archimedean spiral with polar equation

The hyperbolic spiral, also called the inverse spiral (Whittaker 1944, p. 83), originated with Pierre Varignon in 1704 and was studied by Johann Bernoulli between 1710 and 1713, as well as by Cotes in 1722 (MacTutor Archive).

It is also a special case of a Cotes' spiral, i.e., the path followed by a particle in a central orbit with power law

when is a constant and is the specific angular momentum.

The curvature and tangential angle are given by

Fermat's Spiral

Fermat's spiral, also known as the parabolic

spiral, is an Archimedean spiral with

having polar equation

This curve was discussed by Fermat in 1636. For any given positive value of , there are two

corresponding values of of opposite signs. The left plot above shows

only, while the right plot shows equation (1) in red and

in blue. Taking both signs, the resulting spiral is symmetrical about the origin.

Logarithmic Spiral

The logarithmic spiral is a spiral whose polar equation is

given by where is the distance from the origin,

is the angle from the x-axis, and and are arbitrary constants. The logarithmic spiral is also known as the growth spiral, equiangular spiral, and spira

mirabilis. It can be expressed parametrically as

This spiral is related to Fibonacci numbers, the golden ratio, and golden rectangles, and is sometimes called the golden spiral.

The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approached infinity, the sequence of segments approaches the smooth logarithmic spiral (Hilton et al. 1997, pp. 2-3).

The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli. Bernoulli was so fascinated by the spiral that he had one engraved on his tombstone (although the engraver did not draw it true to form) together with the words "eadem mutata resurgo" ("I shall arise the same though changed"). Torricelli worked on it independently and found the length of the curve (MacTutor Archive).

The rate of change of radius is

and the angle between the tangent and radial line at the point is

So, as , and the spiral approaches a circle.

If is any point on the spiral, then the length of the spiral from to the origin is finite. In fact, from

the point which is at distance from the origin measured along a radius vector, the distance from to the pole along the spiral is just the arc length. In addition, any radius from the origin meets the spiral at distances which are in geometric progression (MacTutor Archive).

The arc length (as measured from the origin, ), curvature, and tangential angle of the logarithmic spiral are given by

Conchoid of Nicomedes

A

curve with polar coordinates, , studied by the Greek mathematician Nicomedes in about 200 BC, also known as the cochloid. It is the locus of points a fixed distance away from a line as measured along a line from the focus point. Nicomedes recognized the three distinct forms seen

in this family for , , and . (For , it obviously degenerates to a circle.)

The conchoid of Nicomedes was a favorite with 17th century mathematicians and could be used to solve the problems of cube duplication, angle trisection, heptagon construction, and other Neusis constructions (Johnson 1975).

In Cartesian coordinates, the conchoid of Nicomedes may be written

or

The conchoid has as an asymptote, and the area between either branch and the asymptote is infinite.

A conchoid with has a loop for , where , giving area

The curvature and tangential angle are given by

Conchoid of de Sluze

The

conchoid of de Sluze is the cubic curve first

constructed by René de Sluze in 1662. It is given by the implicit equation

or the polar equation

This can be written in parametric form as

The conchoid of de Sluze has a singular point at the origin which is a crunode for , a cusp

for , and an acnode for .

It has curvature and tangential angle

The curve has a loop if , in which case the loop is swept out by

. The area of the loop is

Cycloid

The cycloid is the locus of a point on the rim of a circle of radius rolling along a straight line. It was studied and named by Galileo in 1599. Galileo attempted to find the area by weighing pieces of metal cut into the shape of the cycloid. Torricelli, Fermat, and Descartes all found the area. The cycloid was also studied by Roberval in 1634, Wren in 1658, Huygens in 1673, and Johann Bernoulli in 1696. Roberval and Wren found the arc length (MacTutor Archive). Gear teeth were also made out of cycloids, as first proposed by Desargues in the 1630s.

In 1696, Johann Bernoulli challenged other mathematicians to find the curve which solves the brachistochrone problem, knowing the solution to be a cycloid. Leibniz, Newton, Jakob Bernoulli and L'Hospital all solved Bernoulli's challenge. The cycloid also solves the tautochrone problem, as alluded to in the following passage from Moby Dick: "[The try-pot] is also a place for profound mathematical meditation. It was in the left-hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along a cycloid, my soapstone, for example, will descend from any point in precisely the same time" (Melville 1851). Because of the frequency with which it provoked quarrels among mathematicians in the 17th century, the cycloid became known as the "Helen of Geometers" (Boyer 1968, p. 389).

The cycloid catacaustic when the rays are parallel to the y-axis is a cycloid with twice as many arches. The radial curve of a cycloid is a circle. The evolute and involute of a cycloid are identical cycloids.

If the cycloid has a cusp at the origin and its humps are oriented upward, its parametric equation is

Humps are completed at values corresponding to successive multiples of , and have height

and length . Eliminating in the above equations gives the Cartesian equation

which is valid for and gives the first half of the first hump of the cycloid. An implicit Cartesian equation is given by

The arc length, curvature, and tangential angle for the first hump of the cycloid are

For the first hump,

For a single hump of the cycloid, the arc length and area under the curve are therefore

Nephroid

The 2-cusped epicycloid is called a nephroid. The name nephroid means "kidney shaped" and was first used for the two-cusped epicycloid by Proctor in 1878 (MacTutor Archive).

The nephroid is the catacaustic for rays originating at the cusp of a cardioid and reflected by it. In addition, Huygens showed in 1678 that the nephroid is the catacaustic of a circle when the light source is at infinity, an observation which he published in his Traité de la luminère in 1690 (MacTutor Archive). (Trott 2004, p. 17, mistakenly states that the catacaustic for parallel light falling on any concave mirror is a nephroid.)

Since the nephroid has cusps, , and the equation for in terms of the parameter

is given by epicycloid equation

with ,

where

This can be written

The parametric equations are

The Cartesian equation is

The nephroid has area and arc length,

The arc length, curvature, and tangential angle as a function of parameter are

where the expressions for and are valid for .

Astroid

A 4-cusped hypocycloid which is sometimes also called a tetracuspid, cubocycloid, or paracycle. The parametric equations of the astroid can be

obtained by plugging in or into the equations for a general

hypocycloid, giving parametric equations

for .

The polar equation can be obtained by computing

and plugging in to to obtain

for .

In Cartesian coordinates,

A generalization of the curve to gives "squashed" astroids, which are a special case of the superellipse

corresponding to parameter .

In pedal coordinates with the pedal point at the center, the equation is

and the Cesàro equation is

A further generalization to an equation of the form

is known as a superellipse.

The arc length, curvature, and tangential angle are

where the formula for holds for .

The perimeter of the entire astroid can be computed from the general hypocycloid formula

with ,

For a squashed astroid, the perimeter has length

The area is given by

with ,

The evolute of an ellipse is a stretched hypocycloid. The gradient of the tangent from the point with

parameter is . The equation of this tangent is

(MacTutor Archive). Let cut the x-axis and the y-axis at and , respectively. Then the length

is a constant and is equal to .

The astroid can also be formed as the envelope produced when a line segment is moved with each end on one of a pair of perpendicular axes (e.g., it is the curve enveloped by a ladder sliding against a wall or a garage door with the top corner moving along a vertical track; left figure above). The

astroid is therefore a glissette. To see this, note that for a ladder of length , the points of contact

with the wall and floor are and , respectively. The equation of the line made

by the ladder with its foot at is therefore

which can be written

The equation of the envelope is given by the simultaneous solution of

which is

Noting that

allows this to be written implicitly as the equation of the astroid.

The related problem obtained by having the "garage

door" of length with an "extension" of length move up and down a slotted track also gives a surprising answer. In this case, the position of the "extended" end for the foot of the door at horizontal

position and angle is given by

Using then gives

Solving ( ) for ◇ , plugging into (◇) and squaring then gives

Rearranging produces the equation

the equation of a (quadrant of an) ellipse with semimajor and

semiminor axes of lengths and .

the astroid is also the envelope of the family of ellipses

Epicycloid

The path traced out by a point on the edge of a circle of radius rolling on the outside of a circle of

radius . An epicycloid is therefore an epitrochoid with .

Epicycloids are given by the parametric equations

A polar equation can be

derived by computing

=

=

so

But

so

Note that is the parameter here, not the polar angle. The polar angle from the center is

To get cusps in the epicycloid, , because then rotations of bring the point on the edge back to its starting position.

so

An epicycloid with one cusp is called a cardioid, one with two cusps is called a nephroid, and one with five cusps is called a ranunculoid.

Epicycloids can also be constructed by beginning with the diameter of a circle and offsetting one end by a series of steps of equal arc length along the circumference while at the same time offsetting the

other end along the circumference by steps times as large. After traveling around the circle

once, the envelope of an -cusped epicycloid is produced, as illustrated above (Madachy 1979).

Epicycloids have torsion

and satisfy

where is the radius of curvature ( ).

Deltoid

A 3-cusped hypocycloid, also called a tricuspoid. The deltoid was first considered by Euler in 1745 in connection with an optical problem. It was also investigated by Steiner in 1856 and is sometimes called Steiner's hypocycloid (Lockwood 1967; Coxeter and Greitzer 1967, p. 44; MacTutor). The equation of the

deltoid is obtained by setting in the equation of the hypocycloid, where is the radius

of the large fixed circle and is the radius of the small rolling circle, yielding the parametric equations

The arc length, curvature, and tangential angle are

The total arc length is computed from the general hypocycloid equation

With , this gives

The area is given by

with

The length of the tangent to the tricuspoid, measured between the two points , in which it cuts

the curve again, is constant and equal to . If you draw tangents at and , they are at right angles.

Rather surprisingly, the deltoid can act as a rotor inside an astroid and, in fact, the deltoid

catacaustic is an astroid.

Ranunculoid

An epicycloid with cusps, named after the buttercup genus Ranunculus (Madachy 1979).

Its parametric equations are

Its arc length and area are given by

Its arc length function, curvature, and tangential angle are

Hypocycloid

The curve produced by fixed point on the circumference of a small circle of radius rolling around

the inside of a large circle of radius . A hypocycloid is therefore a hypotrochoid with .

To derive the equations of the hypocycloid, call the angle by which a point on the small circle rotates

about its center , and the angle from the center of the large circle to that of the small circle . Then

so

Call . If , then the first point is at minimum radius, and the Cartesian parametric equations of the hypocycloid are

If instead so the first point is at maximum radius (on the circle), then the equations of the hypocycloid are

The curvature, arc length, and tangential angle of a hypocycloid are given by

An -cusped hypocycloid has . For an integer and with , the equations of the hypocycloid therefore become

and the arc length and area are therefore

A 2-cusped hypocycloid is a line segment as can be seen by setting in equations (◇) and (◇) and noting that the equations simplify to

This result was noted by the Persian astronomer and mathematician Nasir Al-Din al-Tusi (1201-1274), and is sometimes known as a "Tusi couple" is his honor (Sotiroudis and Paschos 1999, p. 60; Kanas 2003).

The following tables summarizes the names given to this and other hypocycloids with special integer

values of .

hypocycloid

2 line segment (Tusi couple)3 deltoid4 astroid

If is rational, then the curve eventually closes on itself and has cusps. Hypocycloids for a

number of rational values of are illustrated above.

If is irrational, then the curve never closes on itself. Hypocycloids for a number of irrational

values of are illustrated above.

-cusped hypocycloids can also be constructed by beginning with the diameter of a circle, offsetting

one end by a series of steps while at the same time offsetting the other end by steps times as large in the opposite direction and extending beyond the edge of the circle. After traveling around the

circle once, an -cusped hypocycloid is produced, as illustrated above.

Let be the radial distance from a fixed point. For radius of torsion and arc length , a hypocycloid can given by the equation

A hypocycloid also satisfies

where

and is the angle between the radius vector and the tangent to the curve.

The equation of the hypocycloid can be put in a form which is useful in the solution of calculus of

variations problems with radial symmetry. Consider the case , then

But , so , which gives

Now let

so

then

The polar angle is

But

so

Computing

then gives

Finally, plugging back in gives