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In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that + =. This equation , known as the equation of the circle , follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right ...
The general equation for a circle with a center at (,) and radius a is + =. This can be simplified in various ways, to conform to more specific cases, such as the equation r ( φ ) = a {\displaystyle r(\varphi )=a} for a circle with a center at the pole and radius a .
But when looking at a sphere, the integer radius of some adjacent circles is the same, but it is not expected to have the same exact circle adjacent to itself in the same hemisphere. Instead, a circle of the same radius needs a different determinant, to allow the curve to come in slightly closer to the center or extend out farther.
Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is (x − a) 2 + (y − b) 2 = r 2 where a and b are the coordinates of the center (a, b) and r is the radius.
The homothety scales the image by a factor of 2 (a ratio of a diameter to a radius of the sphere), hence the values X and Y produced by this projection are exactly twice those produced by the equatorial projection described in the preceding section. For example, this projection sends the equator to the circle of radius 2 centered at the origin.
The circle with center at Q and with radius R is called the osculating circle to the curve γ at the point P. If C is a regular space curve then the osculating circle is defined in a similar way, using the principal normal vector N. It lies in the osculating plane, the plane spanned by the tangent and principal normal vectors T and N at the ...
Gauss's circle problem asks how many points there are inside this circle of the form (,) where and are both integers. Since the equation of this circle is given in Cartesian coordinates by x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} , the question is equivalently asking how many pairs of integers m and n there are such that
Hence, given the radius, r, center, P c, a point on the circle, P 0 and a unit normal of the plane containing the circle, ^, one parametric equation of the circle starting from the point P 0 and proceeding in a positively oriented (i.e., right-handed) sense about ^ is the following: