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**.

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 = y_{0}, where the subscript 0 denotes the value of the
quantity at some origin. For a circle an equation is s = r.y + s

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**).

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.

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.

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.

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).

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)