Documentation
¶
Overview ¶
Package r3 provides 3D vectors and boxes and operations on them.
Index ¶
- func Cos(p, q Vec) float64
- func Dot(p, q Vec) float64
- func Norm(p Vec) float64
- func Norm2(p Vec) float64
- type Box
- type Mat
- func (m *Mat) Add(a, b mat.Matrix)
- func (m *Mat) At(i, j int) float64
- func (m *Mat) CloneFrom(a mat.Matrix)
- func (m *Mat) Det() float64
- func (m *Mat) Dims() (r, c int)
- func (m *Mat) Mul(a, b mat.Matrix)
- func (m *Mat) MulVec(v Vec) Vec
- func (m *Mat) MulVecTrans(v Vec) Vec
- func (m *Mat) Outer(alpha float64, x, y Vec)
- func (m *Mat) RawMatrix() blas64.General
- func (m *Mat) Scale(f float64, a mat.Matrix)
- func (m *Mat) Set(i, j int, v float64)
- func (m *Mat) Sub(a, b mat.Matrix)
- func (m *Mat) T() mat.Matrix
- func (m *Mat) VecCol(j int) Vec
- func (m *Mat) VecRow(i int) Vec
- type Rotation
- type Vec
Examples ¶
Constants ¶
This section is empty.
Variables ¶
This section is empty.
Functions ¶
Types ¶
type Mat ¶
type Mat struct {
// contains filtered or unexported fields
}
Mat represents a 3×3 matrix. Useful for rotation matrices and such. The zero value is usable as the 3×3 zero matrix.
func NewMat ¶
NewMat returns a new 3×3 matrix Mat type and populates its elements with values passed as argument in row-major form. If val argument is nil then NewMat returns a matrix filled with zeros.
func Skew ¶
Skew returns the 3×3 skew symmetric matrix (right hand system) of v.
⎡ 0 -z y⎤
Skew({x,y,z}) = ⎢ z 0 -x⎥
⎣-y x 0⎦
func (*Mat) Add ¶
Add adds a and b element-wise, placing the result in the receiver. Add will panic if the two matrices do not have the same shape.
func (*Mat) At ¶
At returns the value of a matrix element at row i, column j. At expects indices in the range [0,2]. It will panic if i or j are out of bounds for the matrix.
func (*Mat) CloneFrom ¶
CloneFrom makes a copy of a into the receiver m. Mat expects a 3×3 input matrix.
func (*Mat) Det ¶
Det calculates the determinant of the receiver using the following formula
⎡a b c⎤
m = ⎢d e f⎥
⎣g h i⎦
det(m) = a(ei − fh) − b(di − fg) + c(dh − eg)
func (*Mat) Dims ¶
Dims returns the number of rows and columns of this matrix. This method will always return 3×3 for a Mat.
func (*Mat) Mul ¶
Mul takes the matrix product of a and b, placing the result in the receiver. If the number of columns in a does not equal 3, Mul will panic.
func (*Mat) MulVecTrans ¶
MulVecTrans returns the matrix-vector product Mᵀ⋅v.
func (*Mat) Outer ¶
Outer calculates the outer product of the vectors x and y, where x and y are treated as column vectors, and stores the result in the receiver.
m = alpha * x * yᵀ
func (*Mat) RawMatrix ¶
RawMatrix returns the blas representation of the matrix with the backing data of this matrix. Changes to the returned matrix will be reflected in the receiver.
func (*Mat) Scale ¶
Scale multiplies the elements of a by f, placing the result in the receiver.
See the mat.Scaler interface for more information.
func (*Mat) Sub ¶
Sub subtracts the matrix b from a, placing the result in the receiver. Sub will panic if the two matrices do not have the same shape.
func (*Mat) T ¶
T returns the transpose of Mat. Changes in the receiver will be reflected in the returned matrix.
type Rotation ¶
Rotation describes a rotation in space.
Example (EulerAngles) ¶
package main
import (
"fmt"
"math"
"github.com/gopherd/gonum/num/quat"
"github.com/gopherd/gonum/spatial/r3"
)
// euler returns an r3.Rotation that corresponds to the Euler
// angles alpha, beta and gamma which are rotations around the x,
// y and z axes respectively. The order of rotations is x, y, z;
// there are many conventions for this ordering.
func euler(alpha, beta, gamma float64) r3.Rotation {
// Note that this function can be algebraically simplified
// to reduce floating point operations, but is left in this
// form for clarity.
var rot1, rot2, rot3 quat.Number
rot1.Imag, rot1.Real = math.Sincos(alpha / 2) // x-axis rotation
rot2.Jmag, rot2.Real = math.Sincos(beta / 2) // y-axis rotation
rot3.Kmag, rot3.Real = math.Sincos(gamma / 2) // z-axis rotation
return r3.Rotation(quat.Mul(rot3, quat.Mul(rot2, rot1))) // order of rotations
}
func main() {
// It is possible to interconvert between the quaternion representation
// of a rotation and Euler angles, but this leads to problems.
//
// The first of these is that there are a variety of conventions for
// application of the rotations.
//
// The more serious consequence of using Euler angles is that it is
// possible to put the rotation system into a singularity which results
// in loss of degrees of freedom and so causes gimbal lock. This happens
// when the second axis to be rotated around is rotated to 𝝿/2.
//
// See https://en.wikipedia.org/wiki/Euler_angles for more details.
pt := r3.Vec{1, 0, 0}
// For the Euler conversion function in this example, the second rotation
// is around the y-axis.
const singularY = math.Pi / 2
arb := math.Pi / 4
fmt.Printf("rotate around x-axis: %.2f\n", euler(arb, 0, 0).Rotate(pt))
fmt.Printf("rotate around y-axis: %.2f\n", euler(0, arb, 0).Rotate(pt))
fmt.Printf("rotate around z-axis: %.2f\n", euler(0, 0, arb).Rotate(pt))
fmt.Printf("rotate around x+y-axes: %.2f\n", euler(arb, arb, 0).Rotate(pt))
fmt.Printf("rotate around x+z-axes: %.2f\n", euler(arb, 0, arb).Rotate(pt))
fmt.Printf("rotate around y+z-axes: %.2f\n", euler(0, arb, arb).Rotate(pt))
fmt.Printf("rotate around y-axis to singularity: %.2f\n", euler(0, singularY, 0).Rotate(pt))
fmt.Printf("rotate around x+y-axes with singularity → gimbal lock: %.2f\n", euler(arb, singularY, 0).Rotate(pt))
fmt.Printf("rotate around z+y-axes with singularity → gimbal lock: %.2f\n", euler(0, singularY, arb).Rotate(pt))
fmt.Printf("rotate around all-axes with singularity → gimbal lock: %.2f\n", euler(arb, singularY, arb).Rotate(pt))
}
Output: rotate around x-axis: {1.00 0.00 0.00} rotate around y-axis: {0.71 0.00 -0.71} rotate around z-axis: {0.71 0.71 0.00} rotate around x+y-axes: {0.71 0.00 -0.71} rotate around x+z-axes: {0.71 0.71 0.00} rotate around y+z-axes: {0.50 0.50 -0.71} rotate around y-axis to singularity: {0.00 0.00 -1.00} rotate around x+y-axes with singularity → gimbal lock: {0.00 0.00 -1.00} rotate around z+y-axes with singularity → gimbal lock: {0.00 0.00 -1.00} rotate around all-axes with singularity → gimbal lock: {0.00 0.00 -1.00}
func NewRotation ¶
NewRotation creates a rotation by alpha, around axis.
type Vec ¶
type Vec struct {
X, Y, Z float64
}
Vec is a 3D vector.
func Unit ¶
Unit returns the unit vector colinear to p. Unit returns {NaN,NaN,NaN} for the zero vector.
func (Vec) Rotate ¶
Rotate returns a new vector, rotated by alpha around the provided axis.
DEPRECATED: use r3.Rotate
Source Files