A circle is the locus of points which are equidistant from some fixed point. It is in the plane is determined by a center and a radius. The center is a point in the plane, and the length of the radius is a positive real number, the radius being a line segment from the center to the circumference. The circle consists of all points whose distance from the center equals the radius.
Another way of defining the circle is thus: Given and as two points and as another point, the circle with center is the locus of points with congruent to . (Hilbert, 1927)
Notice there is no definition of distance needed to make that definition and so it works in many geometries, even ones with no distance function. Hilbert uses it in his Foundations of Geometry book Also used by Forder in his Foundations of Euclidean geometry book, c 1927
A circle determines a closed curve in the plane, and this curve is called the perimeter or circumference of the circle. If the radius of a circle is , then the length of the perimeter is . Also, the area of the circle is . More precisely, the interior of the perimeter has area . The diameter of a circle is defined as .
Let us next derive an analytic equation for a circle in Cartesian coordinates . If the circle has center and radius , we obtain the following condition for the points of the sphere,
In other words, the circle is the set of all points that satisfy the above equation. In the special case that , the equation is simply . The unit circle is the circle .
It is clear that equation 1 can always be reduced to the form
where are real numbers. Conversely, suppose that we are given an equation of the above form where are arbitrary real numbers. Next we derive conditions for these constants, so that equation (2) determines a circle . Completing the squares yields
There are three cases:
2 The circle in polar coordinates
Using polar coordinates for the plane, we can parameterize the circle. Consider the circle with center and radius in the plane . It is then natural to introduce polar coordinates for by
with and . Since we wish to parameterize the circle, the point does not pose a problem; it is not part of the circle. Plugging these expressions for into equation (1) yields the condition . The given circle is thus parameterization by , . It follows that a circle is a closed curve in the plane.
3 Three point formula for the circle
Suppose we are given three points on a circle, say , , . We next derive expressions for the parameters in terms of these points. We also derive equation (3), which gives an equation for a circle in terms of a determinant.
First, from equation (2), we have
These equations form a linear set of equations for , i.e.,
Let us denote the matrix on the left hand side by . Also, let us assume that . Then, using Cramer’s rule, we obtain
- 1 D. Hilbert, Foundations of Geometry Chicago: The Open Court Publishing Co. (1921): 163
- 2 J. H. Kindle, Schaum’s Outline Series: Theory and problems of plane of Solid Analytic Geometry, Schaum Publishing Co., 1950.
- 3 E. Weisstein, Eric W. Weisstein’s world of mathematics, http://mathworld.wolfram.com/Circle.htmlentry on the circle.
- 4 L. Råde, B. Westergren, Mathematics Handbook for Science and Engineering, Studentlitteratur, 1995.
|Date of creation||2013-03-22 13:36:23|
|Last modified on||2013-03-22 13:36:23|
|Last modified by||PrimeFan (13766)|
|Defines||three point formula for the circle|