In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a given point, the center. The length of the circle is called its circumference, and any continuous portion of the circle is called an arc.
A circle is a simple closed curve that divides the plane into an interior and exterior. The interior of the circle is called a disk.
Mathematically, a circle can be understood in several other ways as well. For instance, it is a special case of an ellipse in which the two foci coincide (that is, they are the same point). Alternatively, a circle can be thought of as the conic section attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.
All circles have similar properties. Some of these are noted below.
A line segment that connects one point of a circle to another is called a chord. The diameter is a chord that runs through the center of the circle.
In an x-y coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that
If the circle is centered at the origin (0, 0), then this formula can be simplified to
and its tangent will be
where , are the coordinates of the common point.
When expressed in parametric equations, (x, y) can be written using the trigonometric functions sine and cosine as
where t is a parametric variable, understood as the angle the ray to (x, y) makes with the x-axis.
In homogeneous coordinates each conic section with equation of a circle is
It can be proven that a conic section is a circle if and only if the point I(1,i,0) and J(1,-i,0) lie on the conic section. These points are called the circular points at infinity.
In polar coordinates the equation of a circle is
In the complex plane, a circle with a center at c and radius r has the equation . Since , the slightly generalized equation for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles.
The slope of a circle at a point (x, y) can be expressed with the following formula, assuming the center is at the origin and (x, y) is on the circle:
More generally, the slope at a point (x, y) on the circle , (i.e., the circle centered at [a, b] with radius r units), is given by
provided that , of course.
that is, approximately 79 percent of the circumscribed square.
Given that the ratio circumference c to the Area A is
The r and the π can be canceled, leaving
Therefore solving for c:
So the circumference is equal to 2 times the area, divided by the radius. This can be used to calculate the circumference when a value for π cannot be computed.
The diameter of a circle is
An inscribed angle is exactly half of the corresponding central angle (see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle.
Apollonius of Perga showed that a circle may also be defined as the set of points having a constant ratio of distances to two foci, A and B.
The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:
Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to , the angle CPD is exactly , i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.
As a point of clarification, note that C and D are determined by A, B, and the desired ratio (i.e. A and B are not arbitrary points lying on an extension of the diameter of an existing circle).
Given three non-collinear points lying on the circle
The radius of the circle is given by
The center of the circle is given by
A unit normal of the plane containing the circle is given by
Given the radius, , center, , a point on the circle, and a unit normal of the plane containing the circle, , the parametric equation of the circle starting from the point and proceeding counterclockwise is given by the following equation:
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