Spirals, Cycloids, and Generated Curves | Malone Math

The Central Idea

A generated curve is a curve produced by a geometric or mechanical rule of motion.

Instead of asking only

\[ \text{Which points satisfy an equation?} \]

we also ask

\[ \text{How is the point moving to create the curve?} \]

This viewpoint is especially natural for:

spirals;
cycloids and trochoids;
epicycloids and hypocycloids;
involutes and evolutes;
roulettes;
curves created by linkages, rolling, rotation, or constrained motion.

Curves as Traces of Motion

Suppose a moving point has position

\[ \mathbf r(t)=\langle x(t),y(t)\rangle. \]

As \(t\) varies, the point traces a curve.

The formulas

\[ x=x(t),\qquad y=y(t) \]

therefore describe both:

the geometric point set;
the mechanism or order by which the point moves through it.

Why Parametric Form Is Natural for Generated Curves

A generated curve often depends on an angle of rotation, elapsed time, or another continuously changing quantity.

Parametric equations let both coordinates depend on that common quantity:

\[ \boxed{x=x(t),\qquad y=y(t)}. \]

This avoids forcing the curve into a single explicit equation

\[ y=f(x), \]

which may be impossible or geometrically unnatural.

Polar Form Is Natural for Rotational Generation

When the geometry is organized around a central point, polar coordinates are often the most efficient description.

A polar curve has the form

\[ \boxed{r=f(\theta)}. \]

As the angle \(\theta\) changes, the radial distance \(r\) changes according to a rule.

This makes spirals especially natural in polar form.

What Is a Spiral?

A spiral is a curve that winds around a central point while its distance from that point changes.

The central organizing quantities are:

\[ r \]

and

\[ \theta. \]

Different relationships between \(r\) and \(\theta\) generate different spiral families.

The Archimedean Spiral

The Archimedean spiral has polar equation

\[ \boxed{r=a+b\theta}. \]

The simplest form is

\[ \boxed{r=b\theta}. \]

The radius changes linearly with angle.

Equal increases in angle produce equal increases in radial distance.

Spacing of an Archimedean Spiral

After one full revolution,

\[ \theta\mapsto\theta+2\pi. \]

For

\[ r=a+b\theta, \]

the radius changes by

\[ b(2\pi)=2\pi b. \]

Therefore, successive turns of the spiral are separated radially by the constant amount

\[ \boxed{2\pi |b|}. \]

Example: Archimedean Spiral

Consider

\[ r=2\theta. \]

\[ \theta=0, \]

we have

\[ r=0. \]

\[ \theta=\pi, \]

we have

\[ r=2\pi. \]

\[ \theta=2\pi, \]

we have

\[ r=4\pi. \]

The radial distance grows at a constant rate with respect to angle.

The Logarithmic Spiral

A logarithmic spiral has polar equation

\[ \boxed{r=ae^{b\theta}}, \]

where \(a>0\).

Here the radius changes exponentially rather than linearly.

Equal changes in angle multiply the radius by a constant factor.

Scaling Property of a Logarithmic Spiral

Increase the angle by a fixed amount \(\Delta\theta\):

\[ r(\theta+\Delta\theta) =ae^{b(\theta+\Delta\theta)}. \]

Factor:

\[ r(\theta+\Delta\theta) =ae^{b\theta}e^{b\Delta\theta}. \]

Therefore,

\[ \boxed{ \frac{r(\theta+\Delta\theta)}{r(\theta)} =e^{b\Delta\theta} }. \]

The radius changes by a constant ratio for each fixed angular increase.

Archimedean and Logarithmic Spirals Compared

The difference is analogous to linear growth versus exponential growth.

Archimedean spiral:

\[ r=a+b\theta. \]

Logarithmic spiral:

\[ r=ae^{b\theta}. \]

The first has constant radial difference per angular increment. The second has constant radial ratio per angular increment.

The Fermat Spiral

A Fermat spiral can be written

\[ \boxed{r^2=a^2\theta} \]

or equivalently

\[ r=a\sqrt{\theta} \]

on an appropriate domain.

Its radial growth is slower than the linear growth of an Archimedean spiral.

The Hyperbolic Spiral

A hyperbolic spiral can be written

\[ \boxed{r=\frac{a}{\theta}}, \]

where \(\theta\neq 0\).

As \(|\theta|\) grows, the radius approaches zero.

As \(\theta\) approaches zero, the magnitude of \(r\) grows without bound.

Spirals and Orientation

The sign of the angular parameter and the sign of coefficients influence the direction in which a spiral is traced.

Two equations may describe geometrically related spirals while reversing their orientation or reflecting them.

As with other parametric and polar curves, the point set and the tracing rule should be distinguished.

Spirals in Cartesian Parametric Form

Any polar curve

\[ r=f(\theta) \]

can be converted to a Cartesian parametrization:

\[ \boxed{x=f(\theta)\cos\theta}, \] \[ \boxed{y=f(\theta)\sin\theta}. \]

For the Archimedean spiral

\[ r=b\theta, \]

this becomes

\[ \boxed{x=b\theta\cos\theta,\qquad y=b\theta\sin\theta}. \]

What Is a Cycloid?

A cycloid is the path traced by a fixed point on the rim of a circle as the circle rolls along a straight line without slipping.

The curve is generated by combining:

translation of the circle's center;
rotation of the circle;
the fixed offset of the tracing point from the center.

Rolling Without Slipping

Suppose a circle has radius \(a\) and rotates through angle \(t\).

If it rolls without slipping, the distance traveled by its center equals the arc length rolled along the circle:

\[ \boxed{s=at}. \]

Therefore, after rotation through \(t\), the center has moved horizontally by

\[ at. \]

Standard Cycloid Equations

For a circle of radius \(a\) rolling along the \(x\)-axis, one standard cycloid parametrization is

\[ \boxed{x=a(t-\sin t)}, \] \[ \boxed{y=a(1-\cos t)}. \]

The parameter \(t\) is the angle through which the rolling circle has rotated.

Where the Cycloid Formula Comes From

The center of the rolling circle is at

\[ (at,a). \]

Relative to that center, the tracing point contributes

\[ (-a\sin t,-a\cos t). \]

Adding these motions gives

\[ x=at-a\sin t, \] \[ y=a-a\cos t. \]

Hence

\[ \boxed{x=a(t-\sin t),\qquad y=a(1-\cos t)}. \]

Cusps of a Cycloid

The cycloid touches the rolling line whenever the tracing point reaches the bottom of the circle.

This occurs when

\[ t=2\pi k, \]

where \(k\) is an integer.

Then

\[ y=a(1-\cos 2\pi k)=0. \]

These contact points are cusps of the cycloid.

Period of the Cycloid Pattern

One complete rotation changes

\[ t \]

\[ 2\pi. \]

During that rotation, the circle's center advances

\[ 2\pi a. \]

Therefore, successive cycloid arches are horizontally separated by

\[ \boxed{2\pi a}. \]

Highest Point of a Cycloid Arch

The maximum height occurs when

\[ \cos t=-1. \]

This happens at

\[ t=(2k+1)\pi. \]

Then

\[ y=a(1-(-1))=2a. \]

Thus each arch reaches height

\[ \boxed{2a}. \]

A Cycloid Is Not a Sine Wave

The cycloid contains sine and cosine terms, but it is not simply a graph such as

\[ y=\sin x. \]

Both coordinates depend on the parameter:

\[ x=a(t-\sin t), \] \[ y=a(1-\cos t). \]

The horizontal motion is itself nonuniform with respect to \(t\).

Trochoids

A trochoid generalizes the cycloid by allowing the tracing point to lie at distance \(b\) from the center of the rolling circle rather than necessarily on its rim.

One standard parametrization is

\[ \boxed{x=at-b\sin t}, \] \[ \boxed{y=a-b\cos t}. \]

The circle itself still has radius \(a\).

Cycloid as a Special Trochoid

If the tracing distance equals the rolling radius,

\[ b=a, \]

then

\[ x=at-a\sin t, \] \[ y=a-a\cos t, \]

which is exactly the ordinary cycloid.

Therefore,

\[ \boxed{\text{cycloid}=\text{trochoid with }b=a}. \]

Curtate Trochoid

\[ 0 the tracing point lies inside the rolling circle.

The resulting curve is called a curtate trochoid.

It does not develop the sharp cusps of an ordinary cycloid.

Prolate Trochoid

\[ b>a, \]

the tracing point lies outside the rolling circle.

The resulting prolate trochoid develops loops.

The geometry changes because the tracing point extends beyond the rim of the rolling circle.

Rolling on Another Circle

A circle does not have to roll along a straight line.

It can roll around the outside or inside of another circle.

These motions generate several important families:

epicycloids;
hypocycloids;
epitrochoids;
hypotrochoids.

Epicycloid

An epicycloid is traced by a point on a circle that rolls without slipping around the outside of a fixed circle.

If the fixed circle has radius \(R\) and the rolling circle has radius \(r\), a standard parametrization is

\[ \boxed{ x=(R+r)\cos t-r\cos\left(\frac{R+r}{r}t\right) }, \] \[ \boxed{ y=(R+r)\sin t-r\sin\left(\frac{R+r}{r}t\right) }. \]

Hypocycloid

A hypocycloid is traced by a point on a circle that rolls without slipping around the inside of a fixed circle.

One standard parametrization is

\[ \boxed{ x=(R-r)\cos t+r\cos\left(\frac{R-r}{r}t\right) }, \] \[ \boxed{ y=(R-r)\sin t-r\sin\left(\frac{R-r}{r}t\right) }. \]

The Astroid as a Hypocycloid

When the fixed circle has radius four times the rolling radius,

\[ R=4r, \]

the resulting hypocycloid is an astroid.

A common parametrization is

\[ \boxed{x=a\cos^3 t,\qquad y=a\sin^3 t}. \]

Eliminating the parameter gives

\[ \boxed{x^{2/3}+y^{2/3}=a^{2/3}}. \]

Epitrochoids and Hypotrochoids

Epicycloids and hypocycloids use a tracing point on the rolling circle.

If the tracing point is allowed to lie inside or outside that rolling circle, the resulting families are:

epitrochoids for outside rolling;
hypotrochoids for inside rolling.

These families include many looped and rosette-like curves.

Roulettes

A roulette is the path traced by a point attached to a curve as that curve rolls along another curve.

The cycloid is a roulette because a circle rolls along a straight line.

Epicycloids and hypocycloids are also roulettes because one circle rolls along another.

Generated Curves as Composed Motions

Many generated curves can be understood by decomposing motion into pieces.

For a cycloid:

\[ \text{translation}+\text{rotation}. \]

For an epicycloid:

\[ \text{orbital motion}+\text{rotation of the rolling circle}. \]

This decomposition is a geometric version of composing transformations.

Involutes

An involute of a curve can be imagined by unwinding a taut string from the curve while tracking the free end of the string.

The path of the free end forms the involute.

For a circle, the involute has an especially important engineering role in gear-tooth geometry.

Involute of a Circle

For a circle of radius \(a\), one parametrization of an involute is

\[ \boxed{x=a(\cos t+t\sin t)}, \] \[ \boxed{y=a(\sin t-t\cos t)}. \]

The parameter \(t\) measures the angle through which the string has been unwound.

Evolutes

The evolute of a sufficiently smooth curve is the locus of its centers of curvature.

Equivalently, it can be viewed as the envelope of the curve's normal lines under suitable regularity conditions.

Evolutes connect generated geometry with the study of curvature.

Involutes and Evolutes Are Related

Under appropriate smoothness and local conditions, involutes and evolutes are related operations.

Roughly speaking:

\[ \text{an involute unwinds from a curve}, \]

while

\[ \text{an evolute tracks centers of curvature}. \]

The next lesson on curvature will make this relationship more precise.

Curves Generated by Linkages

Mechanical linkages can constrain a point to move along a particular geometric path.

A linkage may convert:

rotation into approximate straight-line motion;
one oscillation into another;
circular motion into a more complicated algebraic curve.

Generated curves therefore connect pure geometry with mechanism design.

Lissajous-Type Generated Curves

Periodic motion in two perpendicular directions can generate curves of the form

\[ \boxed{x=A\sin(at+\delta)}, \] \[ \boxed{y=B\sin(bt)}. \]

These are often called Lissajous curves.

Their shapes depend strongly on the frequency ratio

\[ \frac{a}{b} \]

and the phase difference \(\delta\).

Closed and Nonclosed Generated Curves

A generated curve may close on itself or continue without exactly repeating.

For periodic component motions, closure often depends on whether the relevant frequency ratio is rational.

Rational ratios allow the component motions to return to a common phase after a finite parameter interval.

Generated Curves and Symmetry

The generating mechanism often predicts symmetry before any algebra is performed.

Examples:

a uniformly rolling circle creates repeated translated cycloid arches;
a centered rotational mechanism may create rotational symmetry;
equal-frequency perpendicular oscillations may create reflection symmetry;
inside-circle rolling can create several evenly spaced cusps.

Generated Curves and Periodicity

A curve can repeat geometrically even when it does not return to the same absolute position.

The cycloid, for example, repeats its arch shape after

\[ t\mapsto t+2\pi, \]

but the entire curve is shifted horizontally by

\[ 2\pi a. \]

This is translational repetition rather than closure.

Generated Curves and Scaling

Many generated curves form families controlled by one or more scale parameters.

For a cycloid,

\[ x=a(t-\sin t),\qquad y=a(1-\cos t), \]

multiplying \(a\) by a factor \(k\) scales both coordinates by \(k\).

The entire curve is therefore dilated by that factor.

Generated Curves and Dimensionless Parameters

Parameters such as rotation angles are dimensionless in the usual radian convention.

\[ x=a(t-\sin t), \]

the radius \(a\) carries units of length, while \(t\) and \(\sin t\) are dimensionless.

This keeps the entire right side dimensionally consistent with \(x\).

Generated Curves Can Self-Intersect

Some generated curves loop and cross themselves.

A self-intersection occurs when two different parameter values satisfy

\[ t_1\neq t_2 \]

but

\[ \mathbf r(t_1)=\mathbf r(t_2). \]

Parametric form is especially useful for detecting and interpreting this behavior.

Generated Curves Can Develop Cusps

A cusp is a sharp point at which the regular direction of the curve breaks down.

Cycloids provide a standard example.

\[ t=2\pi k, \]

the tracing point momentarily contacts the rolling line and the cycloid forms a cusp.

Generated Curves and Velocity

\[ \mathbf r(t)=\langle x(t),y(t)\rangle, \]

then the velocity vector is

\[ \boxed{\mathbf v(t)=\mathbf r'(t)}. \]

For the cycloid,

\[ x=a(t-\sin t), \] \[ y=a(1-\cos t), \]

\[ x'(t)=a(1-\cos t), \] \[ y'(t)=a\sin t. \]

Cycloid Speed

The speed is

\[ \sqrt{[x'(t)]^2+[y'(t)]^2}. \]

Substituting gives

\[ a\sqrt{(1-\cos t)^2+\sin^2 t}. \]

Simplify:

\[ (1-\cos t)^2+\sin^2 t =2-2\cos t. \]

Using

\[ 1-\cos t=2\sin^2\frac t2, \]

the speed becomes

\[ \boxed{2a\left|\sin\frac t2\right|}. \]

Why the Cycloid Point Stops at a Cusp

At a cusp,

\[ t=2\pi k. \]

Then

\[ \sin\frac{t}{2}=0. \]

Therefore, the speed of the tracing point is

\[ \boxed{0}. \]

The translation of the wheel and the rotational motion of the rim point cancel instantaneously.

Generated Curves and Arc Length

For a parametric curve,

\[ x=x(t),\qquad y=y(t), \]

the arc length from \(t=\alpha\) to \(t=\beta\) is

\[ \boxed{ L=\int_\alpha^\beta \sqrt{[x'(t)]^2+[y'(t)]^2}\,dt }. \]

Generated curves provide important examples in which a geometric motion leads directly to an analytic length problem.

Arc Length of One Cycloid Arch

For one cycloid arch,

\[ 0\le t\le2\pi. \]

The speed simplifies to

\[ 2a\sin\frac t2 \]

on this interval.

Therefore,

\[ L =\int_0^{2\pi}2a\sin\frac t2\,dt =8a. \]

Thus one complete arch has length

\[ \boxed{8a}. \]

The Brachistochrone Connection

A cycloid appears in the classical brachistochrone problem: finding the curve of fastest descent under uniform gravity between two suitable points.

The solution is an arc of an inverted cycloid, under the standard idealized assumptions of the problem.

This is a striking example of a curve generated by rolling geometry also appearing in optimization and mechanics.

The Tautochrone Connection

A cycloid also appears in the classical tautochrone problem.

Under idealized gravity and frictionless motion, a particle sliding along an inverted cycloidal arc reaches the lowest point in the same amount of time from different starting locations on the arc.

This property links cycloidal geometry with oscillation and dynamics.

Applications of Involute Curves

Involutes of circles are widely used in idealized gear geometry.

Their shape helps maintain a consistent transmission relationship as gear teeth engage.

The application illustrates a general principle:

a curve defined by a geometric generating process can encode useful mechanical behavior.

Computer Generation of Curves

Parametric curves are especially convenient computationally.

Choose parameter values

\[ t_0,t_1,t_2,\ldots \]

and compute

\[ (x(t_i),y(t_i)). \]

A plotting system then connects sufficiently close samples to approximate the curve.

Sampling Resolution Matters

If parameter steps are too large, a computer-generated curve may miss:

small loops;
cusps;
rapid changes in direction;
self-intersections;
fine-scale oscillations.

Numerical representation is therefore an approximation to the continuous curve.

Generated Curves and Coordinate Transformations

Once a curve is generated, it can be translated, rotated, reflected, or scaled.

\[ \mathbf r(t) \]

describes a curve and \(A\) is a transformation matrix, then

\[ \boxed{\mathbf R(t)=A\mathbf r(t)} \]

generates the transformed curve.

Adding a vector translates it:

\[ \boxed{\mathbf R(t)=A\mathbf r(t)+\mathbf b}. \]

Common Error: Confusing the Rolling Radius and Tracing Radius

In the trochoid

\[ x=at-b\sin t, \] \[ y=a-b\cos t, \]

\(a\) is the rolling circle's radius, while \(b\) is the tracing point's distance from the center.

Only when

\[ a=b \]

do we obtain an ordinary cycloid.

Common Error: Treating a Cycloid as a Closed Curve

One arch repeats in shape, but the full cycloid does not close on itself.

Each complete rotation shifts the next arch horizontally by

\[ 2\pi a. \]

The curve is periodic up to translation, not closed.

Common Error: Ignoring Parameter Direction

Replacing \(t\) by \(-t\) can reverse traversal or reflect a curve.

A generated curve should therefore be interpreted using both:

its point set;
the direction in which the mechanism traces it.

Common Error: Assuming All Spirals Have Equal Turn Spacing

The Archimedean spiral has constant radial spacing between successive turns.

The logarithmic spiral does not.

Its radius scales by a constant factor for equal angular increments.

Common Error: Confusing Geometric Periodicity with Coordinate Periodicity

The cycloid's vertical coordinate repeats after \(2\pi\):

\[ y(t+2\pi)=y(t). \]

But

\[ x(t+2\pi)=x(t)+2\pi a. \]

Therefore, the coordinate pair itself is not periodic even though the arch shape repeats.

Common Error: Assuming a Generated Curve Must Have a Simple Cartesian Equation

Parametric and polar equations may be the natural final form.

Eliminating the parameter can produce an equation that is much more complicated or less informative than the generating rule.

A mathematically useful representation does not need to be explicit in \(x\) and \(y\).

A Reliable Procedure for Analyzing a Generated Curve

Identify the generating mechanism.
Choose a parameter tied naturally to the motion.
Write the position of any moving center or reference point.
Add the tracing point's relative displacement.
Simplify the parametric or polar equations.
Determine a useful parameter interval.
Identify symmetry and repetition.
Locate cusps, loops, or self-intersections.
Check scale and dimensional consistency.
Differentiate when velocity, tangent direction, or curvature is needed.

Example: Full Cycloid Analysis

Let a circle of radius \(3\) roll along the \(x\)-axis.

The cycloid is

\[ x=3(t-\sin t), \] \[ y=3(1-\cos t). \]

One arch is generated for

\[ 0\le t\le2\pi. \]

Its horizontal width is

\[ \boxed{6\pi}. \]

Its maximum height is

\[ \boxed{6}. \]

Its cusp points occur at

\[ t=2\pi k. \]

Its one-arch length is

\[ \boxed{24}. \]

Practice 1: Archimedean Spiral

For

\[ r=3\theta, \]

find the radial increase after one complete revolution.

Practice 2: Logarithmic Spiral

For

\[ r=2e^{0.5\theta}, \]

by what factor does the radius change when \(\theta\) increases by \(2\)?

Practice 3: Cycloid Width

A circle of radius \(5\) generates a cycloid.

Find the horizontal width of one complete arch.

Practice 4: Cycloid Height

A cycloid is generated by a circle of radius \(4\).

Find the maximum height of one arch above the rolling line.

Practice 5: Identify the Trochoid Type

A rolling circle has radius

\[ a=6 \]

and the tracing point is

\[ b=2 \]

units from its center.

Is the curve cycloidal, curtate, or prolate?

Practice 6: Identify the Trochoid Type

A rolling circle has radius

\[ a=3 \]

and tracing distance

\[ b=5. \]

Classify the trochoid.

Practice 7: Cycloid Cusp

For

\[ x=2(t-\sin t),\qquad y=2(1-\cos t), \]

find the first two nonnegative parameter values at which cusps occur.

Practice 8: Cycloid Speed

For a cycloid generated by a circle of radius \(a\), find the speed at

\[ t=\pi. \]

Practice 9: Polar to Parametric

Convert

\[ r=2\theta \]

to Cartesian parametric form using \(\theta\) as the parameter.

Practice 10: Identify the Generated Curve

A fixed point on the rim of a circle is traced while the circle rolls without slipping along a straight line.

Name the curve.

Practice Answers

Practice 1

One complete revolution changes \(\theta\) by \(2\pi\).

Therefore,

\[ \Delta r=3(2\pi)=\boxed{6\pi}. \]

Practice 2

Increasing \(\theta\) by \(2\) multiplies the radius by

\[ e^{0.5(2)}=e. \]

Therefore, the factor is

\[ \boxed{e}. \]

Practice 3

One cycloid arch has width

\[ 2\pi a. \]

With \(a=5\),

\[ \boxed{10\pi}. \]

Practice 4

The maximum cycloid height is

\[ 2a. \]

With \(a=4\),

\[ \boxed{8}. \]

Practice 5

Since

\[ b the tracing point lies inside the rolling circle.

The curve is a

\[ \boxed{\text{curtate trochoid}}. \]

Practice 6

Since

\[ b>a, \]

the tracing point lies outside the rolling circle.

The curve is a

\[ \boxed{\text{prolate trochoid}}. \]

Practice 7

Cycloid cusps occur when

\[ t=2\pi k. \]

The first two nonnegative values are

\[ \boxed{0\quad\text{and}\quad2\pi}. \]

Practice 8

Cycloid speed is

\[ 2a\left|\sin\frac t2\right|. \]

At \(t=\pi\),

\[ 2a\left|\sin\frac{\pi}{2}\right| =\boxed{2a}. \]

Practice 9

Use

\[ x=r\cos\theta,\qquad y=r\sin\theta. \]

Since

\[ r=2\theta, \]

we obtain

\[ \boxed{x=2\theta\cos\theta,\qquad y=2\theta\sin\theta}. \]

Practice 10

The generated curve is a

\[ \boxed{\text{cycloid}}. \]

What to Remember

Generated curves are best understood by combining geometry with motion.

An Archimedean spiral has linear radial growth: \[ r=a+b\theta. \]
A logarithmic spiral has exponential radial growth: \[ r=ae^{b\theta}. \]
A cycloid is generated by a point on a circle rolling along a line: \[ x=a(t-\sin t),\qquad y=a(1-\cos t). \]
Trochoids move the tracing point inside or outside the rolling circle.
Epicycloids and hypocycloids arise from one circle rolling around another.
Involutes, evolutes, roulettes, and linkage curves generalize the idea of geometry generated by constrained motion.

The central habit is to ask: what motion or geometric rule creates the path?

Where We Go Next

Generated curves can contain smooth arcs, sharp cusps, loops, and changing directions. To compare these local shapes precisely, we need a way to measure how strongly a curve bends.

The next lesson, Introductory Curvature, develops tangent direction, curvature, radius of curvature, and the local geometry of bending.