Enter the coordinate points into the calculator to determine their inverse points.
Inverse Points Formula
The following equation is used to calculate the Inverse Points.
(x', y') = (R^2 * x / (x^2 + y^2), R^2 * y / (x^2 + y^2))
- Where (x’, y’) is the inverse point
- (x, y) is the original coordinate point
- R is the radius of the circle of inversion
To calculate the inverse points, divide the radius squared by the sum of the squares of the coordinates, and multiply by the original coordinates for both x and y.
What is a Inverse Points?
Definition:
Inverse points refer to points that are transformed with respect to a circle (or another reference) in such a way that their positions become reciprocally related. This is commonly used in geometry to analyze symmetrical properties and reciprocal relationships.
How to Calculate Inverse Points?
Example Problem:
The following example outlines the steps and information needed to calculate the Inverse Points.
First, determine the radius of the circle of inversion. In this example, the circle has a radius of 5.
Next, determine the coordinates of the point. In this case, the point is (4, 3).
Finally, calculate the inverse point using the formula above:
(x’, y’) = (R^2 * x / (x^2 + y^2), R^2 * y / (x^2 + y^2))
(x’, y’) = (5^2 * 4 / (4^2 + 3^2), 5^2 * 3 / (4^2 + 3^2))
(x’, y’) = (25 * 4 / 25, 25 * 3 / 25)
(x’, y’) = (4, 3)
FAQ
What defines the circle of inversion?
The circle of inversion is typically defined by its center and radius. In most cases, the center is taken as the origin (0,0), but translations can shift it elsewhere. The radius determines how far points are reflected or “inverted.”
Can inverse points be used with shapes other than circles?
Inversion is most commonly based on circles. Other shapes can be used in more advanced transformations or generalized forms of inversion, but the classical definition relies on a circle of a given radius.
What happens if a point lies at the center of the circle of inversion?
A point at the center of the circle of inversion does not have a defined inverse, as its distance from the center is zero and division by zero is not possible. In practical applications, this point is typically excluded or addressed separately.