oppositepoint

Oppositepoint is a term used to describe the point that lies opposite a given reference point within a geometric or spatial context. In geometry, it most often refers to the antipodal counterpart on a symmetric object such as a circle or sphere, but the concept can be generalized to other spaces and coordinate systems.

On a circle with center O and radius r, the oppositepoint of a point P on the circle is the point P' such that O is the midpoint of the segment PP'. If the circle is centered at the origin and P has coordinates (x, y) with x^2 + y^2 = r^2, then the oppositepoint is P' = (-x, -y). In a sphere, the oppositepoint (antipode) of a point on the surface is found by extending a line from the center through the point to the opposite surface.

Beyond circles and spheres, oppositepoint can be defined in higher-dimensional spaces or in certain coordinate systems as the point that is collinear with the reference point and the center (or origin) and lies on the opposite side. In some applications, the concept is used in computer graphics, geodesy, and navigation to describe mirror or opposite positions; in these contexts, antipodal point is a more common term.

Relation to antipodal concepts is central, and terminology varies by discipline. Oppositepoint is not a rigorously standardized term in all mathematical texts, but it is often understood as a practical synonym for antipodal points in informal or implementation-specific discussions.