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Circle
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 centre. more...
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Circles are simple closed curves which divide the plane into an interior and exterior. The circumference of a circle means the length of the circle, and the interior of the circle is called a disk. An arc is any continuous portion of a circle.
A circle is a special ellipse in which the two foci coincide (i.e., are the same point). Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.
Analytic results
Equation of a circle
In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that
If the circle is centred at the origin (0, 0), then this formula can be simplified to
and its tangent will be
where x1, y1 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
- ax2 + ay2 + 2b1xz + 2b2yz + cz2 = 0.
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 centre at c and radius r has the equation | z − c | 2 = r2. Since p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles.
Slope
The slope of a circle at a point (x, y) can be expressed with the following formula, assuming the centre is at the origin and (x, y) is on the circle:
Read more at Wikipedia.org
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