This simulator places a fixed point P in world space alongside a movable local origin and its three translated coordinate planes. Adjust the world point and the local origin independently, then watch the projection feet, locating cage and local coordinates update — while the physical point P itself never moves, no matter where the reference origin goes.
• A real-time 3D view of three translucent local XY, XZ and YZ coordinate planes that move with the local origin and divide space into octants, the bright beacon marking world point P, dashed orthogonal projection feet onto each local plane, the local frame translated but kept parallel to the fixed world axes, and a wire locating cage joining the local origin to P — with home view, focus-selected-part, auto-rotate, expand and toggleable labels. • Six live controls: world point x, y and z (each −2 to 2) and local origin x, y and z (each −1 to 1). • Three actions: animate the point moving through space, reset the local origin, and pause the point. • Eight live metrics: local x′, y′ and z′ coordinates, distance to the local origin, distance to the world origin, and distance to each of the local XY, XZ and YZ planes. • A Curves & measurements tab plotting local x′ and y′, the full model equations and snapshot measurements. • An Experiments tab with four guided scenarios (reading an octant, a point on a plane, moving the origin, a coincident origin), a model-verification bench of independent automated checks, and a timestamped event log with a copyable trial report. • A Learn & assess tab with guided lessons, a two-question knowledge-check quiz and a written scope/reference statement.
Locating a point in space always requires three numbers relative to a chosen origin and a set of ordered axes — the signs of those numbers specify which direction along each axis the point lies. When all three local coordinates are nonzero, their sign pattern identifies which of the eight octants the point occupies; setting any single coordinate to zero instead places the point exactly on a coordinate plane.
Moving the local origin does not move the world point P itself — the moving-origin experiment demonstrates this directly by shifting the origin toward a fixed point and showing the local coordinates shrink toward zero while the world coordinates stay exactly what they were. For a translated origin O, the local coordinates are simply P − O: the physical location is invariant, and only the description of that location relative to the reference frame changes.
The equations panel shows P = (x, y, z), O = (ox, oy, oz), the local-coordinate relation P′ = P − O, the distance to the local origin |P − O| = √(x′² + y′² + z′²), and the perpendicular plane distances dXY = |z′|, dXZ = |y′|, dYZ = |x′| — each plane distance depends on exactly the one local coordinate perpendicular to that plane. When a point sits exactly on the local XY plane, its distance to that plane is zero, but the point does not thereby belong to any single octant, since being on a coordinate plane means at least one coordinate sign is undefined as strictly positive or negative.
This model uses two Cartesian frames related purely by translation — there is no rotation, curved coordinates or perspective transformation involved in the coordinate calculation, and the displayed planes are finite patches representing infinite mathematical planes.
No. The world point P stays exactly where it is; only its coordinates relative to the local origin change. Moving the local origin toward P makes the local coordinates shrink, and if the origin is placed exactly on P, all three local coordinates and the origin distance become zero — but P itself has not physically moved anywhere.
It equals |z′|, the absolute value of the local z-coordinate — because the z direction is the one perpendicular to the XY plane. Similarly, distance to the XZ plane equals |y′| and distance to the YZ plane equals |x′|, since each of those coordinates measures displacement perpendicular to its corresponding plane.
An octant is one of the eight regions space is divided into by three coordinate planes, identified by the sign pattern of a point's three local coordinates — for example (+,−,+). A point does not strictly belong to a single octant when one or more of its coordinates is exactly zero, since it then lies on a coordinate plane rather than inside one of the eight open regions.
This model relates two Cartesian frames by translation only — there is no rotation, curved or non-Cartesian coordinates, or perspective transformation in the coordinate calculation itself. The displayed coordinate planes are finite visual patches standing in for planes that are mathematically infinite, and the locating cage is a construction aid connecting the local origin to the point, not a solid volume with any physical significance.