Coordinate Systems (2D)

CoordinateSystem2D

A 2D coordinate system defined by two Direction2D axes and an origin Point2D. Internally it builds a 3×3 affine matrix used for basis-change transforms.

import plane.CoordinateSystem2D
import plane.elements.Direction2D
import plane.elements.Point2D
import units.Angle

// Standard Cartesian system (x right, y up, origin at 0,0)
val standard = CoordinateSystem2D.MAIN_2D_COORDINATE_SYSTEM

// Rotated system — axes tilted 45°
val rotated = CoordinateSystem2D(
    xDirection = Direction2D(1.0, 1.0),   // auto-normalized
    yDirection = Direction2D(-1.0, 1.0),
    origin     = Point2D(2.0, 0.0)
)

Properties

Member	Description
xDirection: Direction2D	x-axis unit direction
yDirection: Direction2D	y-axis unit direction
origin: Point2D	origin of the system
affineMatrix: SimpleMatrix	3×3 affine transform matrix

Rotation

Returns a new CoordinateSystem2D with its axes and origin rotated:

val sys = CoordinateSystem2D.MAIN_2D_COORDINATE_SYSTEM
val rotated = sys.rotate(Angle.Degrees(30.0))

The plot below shows the standard Cartesian system (red/green at origin) alongside a custom 45°-rotated system anchored at (2, 1):

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Changing basis

Any Entity2D can be re-expressed in a different coordinate system using changeBasis():

val systemA = CoordinateSystem2D(
    xDirection = Direction2D(1.0, 0.0),
    yDirection = Direction2D(0.0, 1.0),
    origin     = Point2D(1.0, 0.0)
)

val systemB = CoordinateSystem2D.MAIN_2D_COORDINATE_SYSTEM

val point = Point2D(2.0, 1.0)  // as written in systemA

// Express the same point in systemB
val pointInB = point.changeBasis(asWrittenIn = systemA, to = systemB)

Internally this computes:

systemB.affineMatrix⁻¹ × systemA.affineMatrix × point.affineMatrix

Practical example — expressing a polygon in a tilted frame

import plane.Polygon2D
import plane.CoordinateSystem2D
import plane.elements.Direction2D
import plane.elements.Point2D
import units.Angle

val square = Polygon2D(listOf(
    Point2D(0.0, 0.0),
    Point2D(1.0, 0.0),
    Point2D(1.0, 1.0),
    Point2D(0.0, 1.0)
))

val tiltedSystem = CoordinateSystem2D(
    xDirection = Direction2D(1.0, 1.0),
    yDirection = Direction2D(-1.0, 1.0),
    origin     = Point2D(0.5, 0.5)
)

val squareInTilted = square.changeBasis(
    asWrittenIn = CoordinateSystem2D.MAIN_2D_COORDINATE_SYSTEM,
    to          = tiltedSystem
)