Wed
23
Jan 2013

A class of 2D or 3D point and vector transformations called affine transformations (that is - linear transformation plus translation) can be conveniently represented by matrix. But sometimes we don't need possibility of input x to influence output y or vice versa. For example, that's the case when we want to convert coordinates of mouse cursor from pixels, like we take it from WinAPI (where X goes right from 0 to e.g. 1280 and Y goes down from 0 to e.g. 720) to post-projective 3D space (where X goes right from -1.0 to +1.0 and Y goes up from -1.0 on the bottom to +1.0 on the top). Then every component can be transformed independently using linear transform, like this:

Parameters are: scale.xy, bias.xy.
Given input point i.xy, output point o.xy is:

o.x = i.x * scale.x + bias.x
o.y = i.y * scale.y + bias.y

This seems trivial, but what if we wanted to encapsulate such transformation in a new data structure and define operations on it, like we do on matrices? Let's call it SBX - Scale-Bias Transform. I propose following structure:

struct ScaleBiasXform
{
    static const ScaleBiasXform IDENTITY;
    vec2 Scale;
    vec2 Bias;
};

And following functions:

void Construct(ScaleBiasXform& out,
    const vec2& input1, const vec2& output1,
    const vec2& input2, const vec2& output2);
void Scaling(ScaleBiasXform& out, float scale);
void Scaling(ScaleBiasXform& out, const vec2& scale);
void Translation(ScaleBiasXform& out, const vec2& vec);
void Inverse(ScaleBiasXform& out, const ScaleBiasXform& sbx);
void Combine(ScaleBiasXform& out, const ScaleBiasXform& sbx1, const ScaleBiasXform& sbx2);
void TransformPoint(vec2& out, const vec2& point, const ScaleBiasXform& sbx);
void UntransformPoint(vec2& out, const vec2& point, const ScaleBiasXform& sbx);
void TransformVector(vec2& out, const vec2& vec, const ScaleBiasXform& sbx);
void UntransformVector(vec2& out, const vec2& vec, const ScaleBiasXform& sbx);

Full source code with additional comments and function bodies is here: ScaleBiasTransform.hpp, ScaleBiasTransform.cpp. Some additional notes:

• vec2 is obviously structure containing: float x, y;
• vec2 has some overloaded operators, including lhs*rhs and lhs/rhs, which perform per-component multiplication or division.
• common::CalcLinearFactors used in CPP file is function from my CommonLib, defined in Base.hpp.

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