infinite-copies-of-curve

Infinite Copies of One Curve in Different Positions

Summary: A single equation can encode an entire infinite family of congruent copies of one curve, placed in the plane according to some law (a directrix, a rotation, or both). The technique is to write the curve in body-fixed coordinates $(t,u)$ and substitute the rigid-motion expressions for $t,u$ in terms of laboratory coordinates $(z,y)$ and a varying constant $a$. Examples: circles of fixed radius with vertices on a directrix (figure 90); rotated copies around a center (figure 91); the most general “vertex on directrix + rotating axis” hybrid (figure 92); meridian circles through a fixed point (figure 93).

Sources: chapter18 §§448–456; figures 90–93 in figures90-93.

Last updated: 2026-05-12.

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The setup (§§448–449)

§447 introduces the data: an equation with one varying constant $a$ + variables $x,y$. If the constant appears with the right thread-degree, the resulting curves are similar (similar-curves). §448 notes an alternative: with the same equation, the curves obtained may instead be equal (congruent), differing only in position.

The chapter’s introductory example: $y=a+\sqrt{2cx-x^2}$ is a family of equal circles of radius $c$ whose centers slide along the perpendicular to the axis as $a$ varies (§448).

§449 inverts the question: given the same shape drawn many times in different positions according to some law, find a single equation that captures the family.

1. Translation along a directrix (§449, figure 90)

Take a circle of radius $c$ to be drawn with its vertex sliding along a given curve $AaL$ (the directrix), with the diameter $ab$ kept parallel to the principal axis $AB$.

Construction:

• Let $AK=a$ (abscissa of vertex on the directrix);
• $Ka=A$ — a known function of $a$, determined by the directrix.
• From $a$, draw $ab$ parallel to $AB$ — the diameter of the placed circle.
• For a generic point $m$ of the placed circle, drop perpendicular $mP$ to $AB$ with $AP=z,Pm=y$.

In body-fixed coordinates $ap=t,pm=u$: $t=z-a$, $u=y-A$, and the circle satisfies $u^2=2ct-t^2$. Substituting: $(y-A{)}^2=2c(z-a)-(z-a{)}^2.$ For each value of $a$ (with $A$ a function of $a$ via the directrix), this gives one positioned circle; letting $a$ vary recovers the whole family.

2. Arbitrary curve on a directrix (§450)

The same construction works with any curve whose body-fixed equation is $f(t,u)=0$. Substituting $t=z-a,u=y-A$ where $A=A(a)$ encodes the directrix gives the family equation.

Example: parabola $u^2=ct$ on directrix $AaL$, axes parallel to $AB$, yields $(y-A{)}^2=c(z-a).$

3. Pure rotation about a fixed center (§§451–452, figure 91)

Now keep the vertex on a circle $Aa$ of radius $c$ around center $O$, and let the body’s axis $ab$ always point through $O$. The body-frame transformation is then a rotation by angle $\alpha =\angle AOa$.

With $OP=z,Pm=y$ and angle $\alpha$ varying:

$t=z\cos \alpha +y\sin \alpha -c,u=y\cos \alpha -z\sin \alpha .$

(Derivation: drop $Ps$ parallel to $Ob$ meeting the extended ordinate $mp$ at $s$; then $ps=z\sin \alpha$, $Op-Ps=z\cos \alpha$, $Ps=y\sin \alpha$, $mp=y\cos \alpha -z\sin \alpha$.)

Substituting these into the body equation $f(t,u)=0$ gives one equation in $(z,y)$ whose parameter $\alpha$ traces out all rotated copies.

4. Directrix + rotation combined (§§453–454, figure 92)

The fully general case: vertex on a directrix $AaL$, axis $ab$ tilting through angle $\alpha =\alpha (a)$ as the vertex moves. With $AK=a$, $Ka=A(a)$, $\angle AOa=\alpha (a)$ and from the geometry $KO=A/\tan \alpha$, $Oa=A/\sin \alpha$, Euler derives:

$t=A\sin \alpha -a\cos \alpha +z\cos \alpha -y\sin \alpha ,$ $u=-a\sin \alpha -A\cos \alpha +z\sin \alpha +y\cos \alpha ,$

which can be rewritten compactly as

$t=(z-a)\cos \alpha -(y-A)\sin \alpha ,$ $u=(z-a)\sin \alpha +(y-A)\cos \alpha .$

This is a translation by $(a,A)$ followed by a rotation by $\alpha$ — a general planar rigid motion governed by one parameter $a$. Substituting into the body equation $f(t,u)=0$ packages every positioned copy in one $(z,y)$ equation.

The §455 trichotomy

If the body equation $f(t,u)=0$ has no further constants, the family is of equal (congruent) curves.

If it has a constant that also depends on $a$, and the equation is such that $u$ is a first-degree homogeneous function of $t$ and that constant, the family is of similar curves (recovering similar-curves as a sub-case).

Otherwise the family is of dissimilar curves, but still derivable from one body-frame curve placed in many positions and scales.

§456 worked example: meridian circles (figure 93)

A practical case from cartography: infinite circles all passing through a fixed point $B$ with centers on the line $AE$ — the meridian circles of map projections.

Let $BC=c$ (a fixed perpendicular), $aE=BE=a$ (so circle $amB$ has center $E$ and radius $a$). Then $CE=\sqrt{a^2-c^2}$ and for a generic point $m$ at $CP=z,Pm=y$, the radius condition $P{E}^2+P{m}^2=a^2$ gives:

$y^2+z^2+2z\sqrt{a^2-c^2}-a^2-c^2+a^2=a^2,$ hence $y^2=c^2-2z\sqrt{a^2-c^2}-z^2.$

Letting $CE=a$ (renaming the variable interval) simplifies to $y^2=c^2-2az-z^2,$ the one-parameter family of meridian circles.

The closing remark (§456): “any infinite set of curves which follow some rule can be included in a single equation provided we carefully distinguish between variable and invariable constants.” Chapter 18 ends on this dimensional-bookkeeping principle.

Figures

Figures 90–93

Related pages

• similar-curves — the special case when $u$ is first-degree homogeneous in $t$ and a varying constant.
• affine-curves — the orthogonal variation where scalings rather than positions differ.
• thread-constants — the framework of §455’s trichotomy.
• coordinate-transformations — the rigid-motion substitution used in §§451–454.
• chapter-18-on-the-similarities-and-affinities-of-curves