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Transform and Matrix3d operations

Types

  • A Matrix3d is a 3x3 matrix. It can be used for 3D rotation and scaling.
    • A matrix alone acts on xyz data, but always leaves the origin in place.
    • A matrix must be bundled into a transform (which carries an additional origin/translation term) to cause scaling and rotation around a point other than the origin.
  • A Transform is a 3x3 matrix with an additional point which is variously called "origin" or "translation".

Notation

Indexing for entries in 3x3 matrix >
Indexing for matrix, translation parts of transform >
Typical (column) vector names U,V,W
Typical (row) vector names. (T denotes transpose, which changes the named column to a row) UT, VT, WT
Typical point names A,B,C

Remarks on the entries in a Matrix3d

  • Some common uses of Matrix3d are for:
    • Pure rotation: the matrix "picks something up" and spins it. In the final position, there is not change of distance between any two points marked on the original geometry.
      • the relation of the 9 numbers in the matrix with intuitive description of the rotation is tricky.
      • You cannot pull just a few numbers from the matrix and directly map them to what rotation happened.
    • Uniform scaling: the matrix expands or contracts everything uniformly about the origin.
      • This matrix structure is simple: the scale factor appears on the diagonal. >

Constructor and full update methods

category Matrix3d Transform
create identity newMatrix = Matrix3d.createIdentity () newTransform = Transform.createIdentity ()
matrix.setIdentity () transform.setIdentity ()
all zero newMatrix = Matrix3d.createZero ()
matrix.setZero ()
create new with entries copied from another newMatrix = other.clone () newTransform = other.clone ()
copy values from another existing matrix.setFrom(other) transform.setFrom (other)
create by direct list of numeric entries newMatrix = Matrix3d.createRowValues (qxx, qxy, qxz, qyx, qyy, qyz, qzx, qxy, qzz) > newMatrix = Transform.createRowValues (qxx, qxy, qxz, ax, qyx, qyy, qyz, ay, qzx, qxy, qzz, az) >
Create by rows newMatrix = Matrix3d.createRows (U, V, W) >
Create by columns newMatrix = Matrix3d.createColumns (U, V, W) > newTransform = Transform.createOriginAndMatrixColumns(A,U,V,W) >
Create with columns ordered by an AxisOrder enumeration newMatrix = Matrix3d.createColumnsInAxisOrder (axisOrder, U, V, W) Sample AxisOrder.ZXY places U,V,W as >
set columns matrix.setColumns (U, V, W) transform.setOriginAndMatrixColumns(A,U,V,W)
matrix.setColumn (columnIndex, U)
copy xyz parts from Point4d to columns. matrix.setColumnsPoint4dXYZ (U,V,W)
set row matrix.setRow (rowIndex, U)
create by origin and matrix. newTransform = Transform.createOriginAndMatrix (A, Q) >
create by origin and fixed point. newTransform = Transform.createFixedPointAndMatrix (A, Q) >
uniform scaling in all directions newMatrix = Matrix3d.createUniformScale (s) > Transform.createScaleAboutPoint (fixedPoint, s)
scaling along a single direction newMatrix = Matrix3d.createDirectionalScale (directionVector, scale)
scaling with different factors along each of the 3 axes newMatrix = Matrix3d.createScale (sx, sy, sz) >
rotation around a principal axis newMatrix = Matrix3d.createRotationAxis (axisIndex:AxisIndex, angle)
90 degree rotation around a principal axis newMatrix = Matrix3d.create90DegreeRotationAroundAxis (axisIndex:AxisIndex)
rotation around any vector newMatrix = Matrix3d.createRotationAroundVector (axisVector, angle)
most direct rotation that moves vectorA to vectorB newMatrix = Matrix3d.createRotationVectorToVector (vectorA, vectorB)
rotation that moves vectorA a fraction of the shortest rotation towards vectorB newMatrix = Matrix3d.createPartialRotationVectorToVector (vectorA, fraction, vectorB)

Simple Queries with matrix rows and columns

category Matrix3d
clone individual columns vector = matrix.columnX ()
vector = matrix.columnY ()
vector = matrix.columnZ ()
individual column magnitude a : number = matrix.columnXMagnitude ()
a : number = matrix.columnYMagnitude ()
vector = matrix.columnXMagnitude ()
individual column magnitude squared a : number = matrix.columnXMagnitudeSquared ()
a : number = matrix.columnYMagnitudeSquared ()
a : number = matrix.columnXMagnitudeSquared ()
dot a vector with a specific column a : number = matrix.dotColumnX (vector)
a : number = matrix.dotColumnY (vector)
a : number = matrix.dotColumnZ (vector)
cross a vector with a specific column vector = matrix.columnzZCrossVector(vector)
clone individual rows vector = matrix.rowX ()
vector = matrix.rowY ()
vector = matrix.rowZ ()
individual row magnitude a : number = matrix.rowXMagnitude ()
a : number = matrix.rowYMagnitude ()
a : number = matrix.rowXMagnitude ()
individual row magnitude squared a : number = matrix.rowXMagnitudeSquared ()
a : number = matrix.rowYMagnitudeSquared ()
a : number = matrix.rowXMagnitudeSquared ()
dot a vector with a specific row a : number = matrix.dotRowX (vector)
a : number = matrix.dotRowY (vector)
a : number = matrix.dotRowZ (vector)
dot product within the matrix a: number = matrix.columnXDotColumnY ()
get row or column vector = matrix.getColumn (columnIndex, result?)
vector= matrix.getRow (columnIndex, result)

Rigid Matrix constructions

These methods return individual vectors with special perpendicular conditions related to the input vector and the global axes. These constructions are central to constructing a rigid axis matrix that has a "heads up" sense.

category Matrix3d
return vector perpendicular to input. If input is not near z, the returned vector is in the xy plane newVector = RotMatrix.createPerpendicularVectorFavorXYPlane (zVector)
return a vector which is (a) perpendicular to the input and (b) close to the z axis. newVector = RotMatrix.createHeadsUpPerpendicularNearZ (zVector)

Complex queries

query Input requirements Method
extract axis of rotation Rigit axis matrix rigidMatrix->getAxisAndAngleOfRotation () : {axis: Vector3d, angle: Angle, ok: boolean}

Multiplying points and vectors

method remarks
matrix.mutliplyVector(vector, result)?:Vector3d
matrix.multiplyVectorArrayInPlace (dta: XYZ[])
Matrix3d.XYZMinusMatrixTimesXYZ (origin: XYZ, matrix, vector: XYZ) : Point3d >
Matrix3d.XYZPlusMatrixTimesXYZ (origin: XYZ, matrix, vector: XYZ) : Point3d >
Matrix3d.XYZPlusMatrixTimesCoordinates (origin: XYZ, matrix, x,y,z) : Point3d >
Matrix3d.XYPlusMatrixTimesXY (origin: XAndY, matrix, vector: XAndyY): Point2d >
Matrix3d.XYZPlusMatrixTimesWeightedCoordinates (origin: XAndY, matrix, x,y,z,w): Point4d >
matrix.multiplyTransposeVector (vector): Vector3d >
matrix.multiplyTransposeXYZ (vector): Vector3d >
matrix.multiplyTransposeVectorInPlace (vector): Vector3d >
matrix.multiplyXYZ (x,y,z): Vector3d >
matrix.multiplyVectorInPlace (vector): Vector3d >
multiplyXYZtoXYZ (xyz, result) >
matrix.multiplyXY (x,y): Vector3d >
matrix.originPlusMatrixTimesXY (x,y): Vector3d >

Last Updated: 11 June, 2025