Calculus: Analytic Geometry and Calculus, with Vectors

7.3 Center and radius of curvature 429

most elementaryof elementary geometry tells us that the points
obtained for different values of At would all lie at the center (X,Y)

of the circle. This is no time to be devoid of ideas. We can have at

least a vague feeling that, even when our curve C is not a circle, a small

section of C in a neighborhood of P(t) should be so much like a circle

that there should be a point (X,Y) such that is near (X,Y) whenever

At is small. If all this happens, we should give the point (X,Y) a name

and find formulas for X and Y. All this does happen. The point is

called the center of curvature of C at P(t). We shall find formu'as for

X and Y and for the distance from (X,Y) to P(t). This distance is called

the radius of curvature p (rho) of C at P(t). The circle with center at

(X,Y) and radius p is called the circle of curvature of C at P(t).

The coordinates and n are determined by the system of equations

[ - x(t)]x'(t) + [n - y(t)]y'(t) = 0

[ - x(t + At)]x'(t + At) + [n - y(t + At)]y'(t + At) = 0.

The left member of the first equation is the scalar product of the vector

v(t) tangent to C at P(t) and the vector running from P(t) to %,J), and

the equation expresses the fact that the two vectors are orthogonal.

Similarly, the second equation expresses the fact that (g,n) lies on the

normal to C at P(t + At). Replacing the quantities in brackets in the

second equation by

[ - x(t) - x(t + At) + x(t)] and In - y(t) - y(t + At) + y(t)]

and transposing a part of the result enables us to put the two equations

in the form

(7.341) [ - x(t)]x'(t) + [n - y(t)]y'(t) = 0

(7.342) [ - x(t)]x'(t + At) + [n - y(t)ly'(t + At) = Q,

where we have simplified matters by letting Q denote the quantity

(7.343) Q = [x(t + At) - x(t)]x'(t + At) + [y(t + At) - y(t)]y'(t + At).

To eliminate n from the two equations (7.341) and (7.342), we multiply

the first by y'(t + At) and the second by -y(t) and add to obtain the

first of the formulas

(7.35) D[ - x(t)] = -Qy'(t), D[n - y(t)] = Qx'(t),

where D is the determinant

D = x (t)y'(t + At) - x'(t + At)y'(t)

which can be put in the form

(7.351) D = x'(t)[y'(t + At) - y'(t)l - [x'(t + At) - x'(t)ly'(t)