A straight line segment between two points on a curve.
circle
A curved planar figure, all points of which are the same
distance from a fixed point called the center.
diameter
1) A line segment connecting two points on a circle, and
passing through its center. 2) Twice the radius (of a
circle).
radius
1) A line segment from a circle to its center. 2)The
distance from any point on a circle to its center.
Geometric Definition
Geometrically, a circle can be defined as:
Given a fixed point C, a circle is the figure formed from all
points P for which PC is constant.
The fixed point is called the center.
135
Analytic Definition
Analytically, a circle can be defined as:
Given a constant r, a circle is the locus of all points
(x, y) for which
x^{2}
y^{2}
-----
+
-----
=
1
r^{2}
r^{2}
or
x^{2} + y^{2} = r^{2}
Analysis
The geometric definition and the analytic definition are
equivalent, and will produce the same figure for center
(0, 0)
Line
Segments
A line segment from (x, y) to (0, 0), is called a
radius of the circle.
A line segment from (x, y) to (-x, -y), is called a
diameter of the circle.
These terms are also used to refer to the lengths of these line
segments.
A line segment from (x_{1}, y_{1}) to
(x_{2}, y_{2}) is called a chord of the
circle.
Intercepts
A circle (in the standard form) intercepts the x axis at
(r, 0) and (-r, 0).
A circle (in the standard form) intercepts the y axis at
(0, r) and (0, -r).
Excentricity
The excentricity e of a circle is
0
e
=
----
=
0
a
For all circles, e = 0;
General Case
The equation
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0
will describe a circle if the discriminate,
B^{2} - 4AC,
is equal to 0.
Parametric Definition
A circle can also be defined parametrically:
Given a constant r and a parameter θ, a circle is the
locus of all points (x, y) for which
x = r cos θ, and
y = r sin θ
Polar Definition
A circle can also be defined in polar coordinates
Given a constant r greater than 0, a circle is the locus of
all points (ρ, θ) for which