Fig. 81.

The Ovals Of Cassini

272. When a point moves in a plane so that the product of its distances from two fixed points in the plane is constant, it traces out one of Cassini's ovals. The fixed points are called the foci. The equation of the curve is rr' = k2, where r and r' are the distances of any point on the curve from the foci and k is a constant.

Fig. 82.

Let F and F' be the foci. Fold through F and F'. Bisect FF' in C, and fold BCB' perpendicular to FF'. Find points B and B' such that FB and FB' are each =k. Then B and B' are evidently points on the curve.

Fold FK perpendicular to FF' and make FK=k, and on FF' take CA and CA' each equal to CK. Then A and A' are points on the curve.

For CA2 =CK2= CF2 + FK2

... CA2 - CF2=k2 = (CA+CF)(CA - CF)

= F'A.FA.

Produce FA and take AT=FK. In AT take a point M and draw MK. Fold KM' perpendicular to MK meeting FA' in M'.

Then FM.FM' = k2.

With the center F and radius FM, and with the center F' and radius FM', describe two arcs cutting each other in P. Then P is a point on the curve.

When a number of points between A and B are found, corresponding points in the other quadrants can be marked by paper folding.

When FF' = √2k and rr' = ½k2 the curve assumes the form of a lemniscate. (§ 279.)

When FF' is greater than √2k, the curve consists of two distinct ovals, one about each focus.

The Logarithmic Curve

273. The equation to this curve is y = ax.

The ordinate at the origin is unity.

If the abscissa increases arithmetically, the ordinate increases geometrically.

The values of y for integral values of x can be obtained by the process given in § 108.

The curve extends to infinity in the angular space XOY.

If x be negative y = 1 / ax and approaches zero as x increases numerically. The negative side of the axis OX is therefore an asymptote to the curve.

The Common Catenary

274. The catenary is the form assumed by a heavy inextensible string freely suspended from two points and hanging under the action of gravity.

The equation of the curve is c the axis of y being a vertical line through the lowest point of the curve, and the axis of x a horizontal line in the plane of the string at a distance c below the lowest point; c is the parameter of the curve, and e the base of the natural system of logarithms.

When x = c, y = c/2 (e1 + e-1) when x = 2c, y= c/2 (e2 + e-2) and so on.

275. From the equation e can be determined graphically.

ce - 2y√e + c = 0

√e = 1/c (y + c√ e=y+

is found by taking the geometric mean between y + c and y - c.

The Cardioid Or Heart-Shaped Curve

276. From a fixed point O on a circle of radius a draw a pencil of lines and take off on each ray, measured both ways from the circumference, a segment equal to 2a. The ends of these lines lie on a cardioid.

Fig. 83.

The equation to the curve is r = a(1 + cosθ).

The origin is a cusp on the curve. The cardioid is the inverse of the parabola with reference to its focus as center of inversion.

The Limacon

277. From a fixed point on a circle, draw a number of chords, and take off a constant length on each of these lines measured both ways from the circumference of the circle.

If the constant length is equal to the diameter of the circle, the curve is a cardioid.

If it be greater than the diameter, the curve is altogether outside the circle.

If it be less than the diameter, a portion of the curve lies inside the circle in the form of a loop.

If the constant length is exactly half the diameter, the curve is called the trisectrix, since by its aid any angle can be trisected.

The equation is r = a cos θ + b.

Fig. 84.

The first sort of limacon is the inverse of an ellipse; and the second sort is the inverse of an hyperbola, with reference to a focus as a center. The loop is the inverse of the branch about the other focus.

278. The trisectrix is applied as follows: Let A OB be the given angle. Take OA, OB equal to the radius of the circle. Describe a circle with the center 0 and radius OA or OB. Produce AO indefinitely beyond the circle. Apply the trisectrix so that O may correspond to the center of the circle and OB the axis of the loop. Let the outer curve cut AO produced in C. Draw BC cutting the circle in D, Draw OD.

Fig. 85.

Then ACB is 1/3 ofA OB.

For CD = DO = OB.

... AOB =ACB+CBO =ACB+ODB =ACB+2ACS = 3ACB.

The Lemniscate Of Bernoulli

279. The polar equation to the curve is r2 = a2cos2θ. Let 0 be the origin, and OA=a. Produce AO, and draw OD at right angles to OA Take the angle AOP =θ and AOB =2θ. Draw AB perpendicular to OB. In AO produced take OC=OB.

Find D in OD such that CD A is a right angle.

Take OP = OD.

P is a point on the curve.

r2=OD2 = OC.OA = OB.OA = a cos 2θ .a = a2 cos2θ. As stated above, this curve is a particular case of the ovals of Cassini.

Fig. 86.

It is the inverse of the rectangular hyperbola, with reference to its center as center of inversion, and also its pedal with respect to the center.

The area of the curve is a2.

The Cycloid

280. The cycloid is the path described by a point on the circumference of a circle which is supposed to roll upon a fixed straight line.

Let A and A' be the positions of the generating point when in contact with the fixed line after one complete revolution of the circle. Then AA' is equal to the circumference of the circle.

The circumference of a circle may be obtained in length in this way. Wrap a strip of paper round a circular object, e. g., the cylinder in Kindergarten gift No. II., and mark off two coincident points. Unfold the paper and fold through the points. Then the straight line between the two points is equal to the circumference corresponding to the diameter of the cylinder.

By proportion, the circumference corresponding to any diameter can be found and vice versa.

Fig. 87.

Bisect AA' in D and draw DB at right angles to AA', and equal to the diameter of the generating circle.

Then A, A' and B are points on the curve.

Find O the middle point of BD.

Fold a number of radii of the generating circle through O dividing the semi-circumference to the right into equal arcs, say, four.

Divide AD into the same number of equal parts.

Through the ends of the diameters fold lines at right angles to BD.

Let EFP be one of these lines, F being the end of a radius, and let G be the corresponding point of section of AD, commencing from D. Mark off FP equal to GA or to the length of arc BF.

Then P is a point on the curve.

Other points corresponding to other points of section of AD may be marked in the same way.

The curve is symmetric to the axis BD and corresponding points on the other half of the curve can be marked by folding on BD.

The length of the curve is 4 times BD and its area 3 times the area of the generating circle.

The Trochoid

281. If as in the cycloid, a circle rolls along a straight line, any point in the plane of the circle but not on its circumference traces out the curve called a trochoid.

The Epicycloid

282. An epicycloid is the path described by a point on the circumference of a circle which rolls on the circumference of another fixed circle touching it on the outside.

The Hypocycloid

283. If the rolling circle touches the inside of the fixed circle, the curve traced by a point on the circumference of the former is a hypocycloid.

When the radius of the rolling circle is a sub-multiple of the fixed circle, the circumference of the latter has to be divided in the same ratio.

These sections being divided into a number of equal parts, the position of the center of the rolling circle and of the generating point corresponding to each point of section of the fixed circle can be found by dividing the circumference of the rolling circle into the same number of equal parts.

The Quadratrix

284. Let OACB be a square. If the radius OA of a circle rotate uniformly round the center O from the position OA through a right angle to OB and if in the same time a straight line drawn perpendicular to OB move uniformly parallel to itself from the position OA to BC; the locus of their intersection will be the quadratrix.

This curve was invented by Hippias of Elis (420 B. C.) for the multisection of an angle.

If P and P' are points on the curve, the angles A OP ana A OP' are to one another as the ordinates of the respective points.

* Beman and Smith's translation of Klein's Famous Problems of Elementary Geometry, p. 57.

The Spiral Of Archimedes

285. If the line OA revolve uniformly round O as center, while point P moves uniformly from O along OA, then the point P will describe the spiral of Archimedes.