Curves

Curved and Straight Lines

A straight line is called a tangent to a curve if it touches the curve (i.e. has a point in common with the curve) but does not actually cross the line of the curve at that point. There may be more than one tangent at a given point of a curve, in which case that point is called a cusp of the curve. If there is one tangent to a curve at a given point of the curve then the slope of the tangent line is called the slope of the curve at that point. A curve with a tangent at every point is called a smooth curve. A line perpendicular to a tangent at the point of contact of the tangent is called a normal to the curve.

A straight line cutting a curve in two or more points is called a secant, and a straight line segment cut off between two points on a curve is called a chord.

Curvature

If y (psi) denotes the angle through which the tangent to a smooth curve turns in an arc length s, then y/s is the (mean) curvature of that section of the curve. If we can determine the curvature of a curve at any point on it we call it well-behaved, and identify the curvature with the rate of change of y with respect to s, which we write y ~ s. In the case of a straight line y = 0 for all values of s, so its curvature is zero at every point. In the case of a circle of radius r we have s = r.y, by definition of angle, so y/s = 1/r. Thus a circle is a curve of constant, non-zero, curvature and the curvature is measured by the reciprocal of the radius. Larger circles have smaller curvature.

A circle tangent to a curve and having the same curvature as the curve at the point of contact is called the circle of curvature, its radius the radius of curvature, and its centre the centre of curvature of the curve at that point. The point of intersection of the two normals to the curve at the endpoints of the arc through the point approaches the centre of curvature as the arc-length is reduced. A point of the curve at which the curvature is momentarily zero is called a point of inflexion of the curve. At this point there is no circle of curvature but there is a straight line that has the same slope and curvature at that point (and is called a tangent, though it does not quite fit the definition of tangent given above). At a cusp the curvature becomes infinite and the circle of curvature reduces to a point.

The curvature y ~ s can most readily be calculated if the relation between y and s for the curve, for some choice of origins, is known. Such a relation is called an intrinsic equation for the curve. For a straight line an equation is y = y0, where the subscript 0 denotes the value of the quantity at some origin. For a circle an equation is s = r.y + s, with r constant.

The locus of the centre of curvature is called the evolute of a curve. The normals to the curve are the tangents to the evolute. Conversely the evolute is the envelope of the normals. It may be possible to draw the curve by attaching a pen to the end of a string stretched over the evolute (a curve drawn in this way is called an involute).


Ovals

A convex closed smooth curve may be termed an oval. By convex we mean that any straight line segment joining two points on the curve is entirely within the curve. From any point outside an oval there are two tangents to the curve.

Every tangent will have a parallel tangent opposite, and a line joining the points of contact of two such tangents is a diameter of the oval. As the tangents move round the curve their separation varies continuously, reaching maximum and minimum values and all values between them. If in the cycle of values there is one maximum and one minimum then the curve may be called a simple oval and the minimum and maximum diameters its major and minor diameters, and their lengths the length and breadth of the oval. If there is more than one maximum or minimum the curve is a complex oval.

There will be another pair of tangents perpendicular to a given pair of parallel tangents, thus enclosing the oval inside a rectangle. As the tangents rotate the rectangle varies in shape continuously, each side reaching the same maxima and minima but 90° out of phase. So at some point the rectangle must become square. Thus every oval can be inscribed in a square, touching all four sides.

The locus of the vertices of the enclosing rectangle is a larger oval, surrounding the given one. It is the locus of points from which the tangents to the oval are at right angles.


Circles

The set of all points at a given distance from a given point constitutes a circle, which is a closed curve, enclosing a circular region. The given point is the centre and the given distance is the radius r. Any line segment from centre to circle is also called a radius. The length of the curve, the circumference of the region, is 2.p.r.

A secant through the centre is a diameter, and divides the circle into two equal arcs called semicircles. Every diameter is an axis of symmetry. Any other chord divides the circle into two arcs of different lengths, greater and smaller than a semicircle, called major and minor arcs. All angles subtended by a chord at a point on an arc are equal. The angle in the major arc is acute and in the minor arc is obtuse, and their sum is 180°. The angle in the major arc is half the angle subtended by the same chord at the centre of the circle.

A tangent is perpendicular to the radius at the point of contact. From any point outside the circle the two tangents that can be drawn to the circle are equal in length, and their angle is bisected by the diameter through the same point. The chord connecting the two points of contact of the tangents is at right angles to the diameter and bisected by it. The minor angle subtended by a chord is equal to the angle between the chord and the tangent at one end of the chord.


Piece-wise Circular Curves

The figure illustrates an exercise with compass and rule for constructing ovals. The basic idea is to form a smooth curve by joining several circular arcs of different radii. There must be no sharp bend (i.e. angle) in the curve at the point where the two arcs meet. A straight line through the centres of the two arcs must pass through the point where they join. The radii can be any pair of sizes and can even point in opposite directions. Any curve can be approximated by a number of circular arcs, or indeed straight lines.


Curves of constant width (defined as the distance between parallel tangents) can be constructed by this method. A regular triangle with its sides replaced by circular arcs, centred on its vertices has this property but it has points at the corners where the gradient is discontinuous. Using a regular triangle with sides extended slightly, six circular arcs centred on the vertices will now give a smooth curve. This method can in fact be applied to any triangle, regular or not, and to any polygon with an odd number of sides. (For instance the seven-sided 50p coin.) If the width is d the length of the curve is p.d, as for a circle.

Maxwell's Ovals

James Clerk Maxwell in a youthful paper considered the drawing of ovals with more than two foci such that the sum of the three distances from the foci is constant (r + s + t = k). His method of drawing the ovals is to attach to one focus a string of length k + d, where k is the required sum of the three distances and d is the distance between the other two foci. The string then passes round the pen, then round the other two foci and back to the pen (see dotted line).

The example shows foci at the vertices of a 3,4,5 triangle. (Drawing very rough).


Astroid and Semicubical Ellipses

The astroid is the 4-cusped hypocycloid formed by rolling a circle on the inside of a larger circle of 4 times the diameter. The algebraic equation of the astroid is x^(2/3) + y^(2/3) = a^(2/3).

A quarter of it is also the envelope of all the positions of a ladder placed to lean against a vertical wall while standing on a horizontal floor. The locus of the centre point of the ladder is a circular arc. Since the instantaneous centre for the ladder's motion is at I, on the circle of radius a, where a is the length of the ladder, the point of contact of the ladder with the envelope is at N, the foot of the perpendicular from I to the ladder.


A semicubical ellipse may be obtained from an astroid by orthogonal projection (i.e. squashing or stretching), in the same way that an elipse is obtained from a circle.

It is also the evolute of an ellipse. The normal at any point of an ellipse x = a.cosq, y = b.sinq is given by a.x/cosq − b.y/sinq = a² − b². Differentiating with respect to q we find a.x/(cosq)³ − b.y/(sinq)³ = a² − b². Hence the coordinates of the centre of curvature are x = (a² − b²).(cosq)³/a, y = (a² − b²).(sinq)³/b and the evolute is (a.x)^(2/3) + (b.y)^(2/3) = (a² − b²)^(2/3). The centres of curvature at the points A, A', B, B' are E, E', F, F' respectively. (Rough drawing only)