# Shear6¶

```#include <Imath/ImathShear.h>
```

The `Shear6` class template represent a 3D shear transformation, with predefined typedefs for `float` and `double`.

Example:

```#include <Imath/ImathMatrix.h>
#include <Imath/ImathShear.h>

void
shear6_example ()
{
Imath::Shear6f s (0.330f, 0.710f, 0.010f, 0.999f, -0.531f, -0.012f);

Imath::M44f M;
M.setShear (s);
}
```
typedef Shear6<float> Imath::Shear6f

Shear6 of type float.

template<class T>
class Shear6

Shear6 class template.

A shear matrix is technically defined as having a single nonzero off-diagonal element; more generally, a shear transformation is defined by those off-diagonal elements, so in 3D, that means there are 6 possible elements/coefficients:

```| X' |   |  1  YX  ZX  0 |   | X |
| Y' |   | XY   1  ZY  0 |   | Y |
| Z' | = | XZ  YZ   1  0 | = | Z |
| 1  |   |  0   0   0  1 |   | 1 |

X' =      X + YX * Y + ZX * Z
Y' = YX * X +      Y + ZY * Z
Z` = XZ * X + YZ * Y +      Z
```
See https://www.cs.drexel.edu/~david/Classes/CS430/Lectures/L-04_3DTransformations.6.pdf

Those variable elements correspond to the 6 values in a Shear6. So, looking at those equations, “Shear YX”, for example, means that for any point transformed by that matrix, its X values will have some of their Y values added. If you’re talking about “Axis A has values from Axis B added to it”, there are 6 permutations for A and B (XY, XZ, YX, YZ, ZX, ZY).

Not that Maya has only three values, which represent the lower/upper (depending on column/row major) triangle of the matrix. Houdini is the same as Maya (see https://www.sidefx.com/docs/houdini/props/obj.html) in this respect.

There’s another way to look at it. A general affine transformation in 3D has 12 degrees of freedom - 12 “available” elements in the 4x4 matrix since a single row/column must be (0,0,0,1). If you add up the degrees of freedom from Maya:

• 3 translation

• 3 rotation

• 3 scale

• 3 shear

You obviously get the full 12. So technically, the Shear6 option of having all 6 shear options is overkill; Imath/Shear6 has 15 values for a 12-degree-of-freedom transformation. This means that any nonzero values in those last 3 shear coefficients can be represented in those standard 12 degrees of freedom. Here’s a python example of how to do that:

```>>> import imath
>>> M = imath.M44f()
>>> s = imath.V3f()
>>> h = imath.V3f()
>>> r = imath.V3f()
>>> t = imath.V3f()
# Use Shear.YX (index 3), which is an "extra" shear value
>>> M.setShear((0,0,0,1,0,0))
M44f((1, 1, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1))
>>> M.extractSHRT(s, h, r, t)
1
>>> s
V3f(1.41421354, 0.707106769, 1)
>>> h
V3f(1, 0, 0)
>>> r
V3f(0, -0, 0.785398185)
>>> t
V3f(0, 0, 0)
```
That shows how to decompose a transform matrix with one of those “extra” shear coefficients into those standard 12 degrees of freedom. But it’s not necessarily intuitive; in this case, a single non-zero shear coefficient resulted in a transform that has non-uniform scale, a single “standard” shear value, and some rotation.

So, it would seem that any transform with those extra shear values set could be translated into Maya to produce the exact same transformation matrix; but doing this is probably pretty undesirable, since the result would have some surprising values on the other transformation attributes, despite being technically correct.

This usage of “degrees of freedom” is a bit hand-wavey here; having a total of 12 inputs into the construction of a standard transformation matrix doesn’t necessarily mean that the matrix has 12 true degrees of freedom, but the standard translation/rotation/scale/shear matrices have the right construction to ensure that.

T xy
T xz
T yz
T yx
T zx
T zy

Constructors and Assignment

inline constexpr Shear6()

Initialize to 0.

inline constexpr Shear6(T XY, T XZ, T YZ)

Initialize to the given XY, XZ, YZ values.

inline constexpr Shear6(const Vec3<T> &v)

Initialize to the given XY, XZ, YZ values held in (v.x, v.y, v.z)

template<class S>
inline constexpr Shear6(const Vec3<S> &v)

Initialize to the given XY, XZ, YZ values held in (v.x, v.y, v.z)

inline constexpr Shear6(T XY, T XZ, T YZ, T YX, T ZX, T ZY)

Initialize to the given (XY XZ YZ YX ZX ZY) values.

inline constexpr Shear6(const Shear6 &h)

Copy constructor.

template<class S>
inline constexpr Shear6(const Shear6<S> &h)

Construct from a Shear6 object of another base type.

inline constexpr const Shear6 &operator=(const Shear6 &h)

Assignment.

template<class S>
constexpr const Shear6 &operator=(const Vec3<S> &v)

Assignment from vector.

~Shear6() = default

Destructor.

Compatibility with Sb

template<class S>
inline void setValue(S XY, S XZ, S YZ, S YX, S ZX, S ZY)

Set the value.

template<class S>
inline void setValue(const Shear6<S> &h)

Set the value.

template<class S>
inline void getValue(S &XY, S &XZ, S &YZ, S &YX, S &ZX, S &ZY) const

Return the values.

template<class S>
inline void getValue(Shear6<S> &h) const

Return the value in `h`

inline T *getValue()

Return a raw pointer to the array of values.

inline const T *getValue() const

Return a raw pointer to the array of values.

Arithmetic and Comparison

template<class S>
inline constexpr bool operator==(const Shear6<S> &h) const

Equality.

template<class S>
inline constexpr bool operator!=(const Shear6<S> &h) const

Inequality.

inline constexpr bool equalWithAbsError(const Shear6<T> &h, T e) const

Compare two shears and test if they are “approximately equal”:

Returns:

True if the coefficients of this and h are the same with an absolute error of no more than e, i.e., for all i abs (this[i] - h[i]) <= e

inline constexpr bool equalWithRelError(const Shear6<T> &h, T e) const

Compare two shears and test if they are “approximately equal”:

Returns:

True if the coefficients of this and h are the same with a relative error of no more than e, i.e., for all i abs (this[i] - h[i]) <= e * abs (this[i])

inline constexpr const Shear6 &operator+=(const Shear6 &h)

inline constexpr Shear6 operator+(const Shear6 &h) const

inline constexpr const Shear6 &operator-=(const Shear6 &h)

Component-wise subtraction.

inline constexpr Shear6 operator-(const Shear6 &h) const

Component-wise subtraction.

inline constexpr Shear6 operator-() const

Component-wise multiplication by -1.

inline constexpr const Shear6 &negate()

Component-wise multiplication by -1.

inline constexpr const Shear6 &operator*=(const Shear6 &h)

Component-wise multiplication.

inline constexpr const Shear6 &operator*=(T a)

Scalar multiplication.

inline constexpr Shear6 operator*(const Shear6 &h) const

Component-wise multiplication.

inline constexpr Shear6 operator*(T a) const

Scalar multiplication.

inline constexpr const Shear6 &operator/=(const Shear6 &h)

Component-wise division.

inline constexpr const Shear6 &operator/=(T a)

Scalar division.

inline constexpr Shear6 operator/(const Shear6 &h) const

Component-wise division.

inline constexpr Shear6 operator/(T a) const

Scalar division.

Numerical Limits

static inline constexpr T baseTypeLowest() noexcept

Largest possible negative value.

static inline constexpr T baseTypeMax() noexcept

Largest possible positive value.

static inline constexpr T baseTypeSmallest() noexcept

Smallest possible positive value.

static inline constexpr T baseTypeEpsilon() noexcept

Smallest possible e for which 1+e != 1.

Public Types

typedef T BaseType

The base type: In templates that accept a parameter `V` (could be a Color4), you can refer to `T` as `V::BaseType`

Public Functions

inline constexpr T &operator[](int i)

Element access.

inline constexpr const T &operator[](int i) const

Element access.

template<class S>
inline constexpr const Shear6<T> &operator=(const Vec3<S> &v)

Public Static Functions

static inline constexpr unsigned int dimensions()

Return the number of dimensions, i.e. 6.

template<class T>
std::ostream &Imath::operator<<(std::ostream &s, const Shear6<T> &h)

Stream output, as “(xy xz yz yx zx zy)”.