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Biquadratic polynomial

As explained in Geometry for Programmers

A biquadratic polynomial is a two-variable polynomial with degree at most two in each variable.

A biquadratic polynomial uses two variables, each appearing with degrees 0, 1, or 2. Its nine terms form a 3 × 3 grid of combinations, from x²y² through the constant term. section 8.1.1

Pairing two such polynomials produces a biquadratic transformation: one maps input coordinates to xᵗ, and the other maps them to yᵗ. Nine corresponding point pairs determine the coefficients. section 8.1.1

Coordinate transformation

A transformed x-coordinate can be written as xᵗ = a₁₁x²y² + a₁₂x²y + a₁₃x² + a₁₄xy² + a₁₅xy + a₁₆x + a₁₇y² + a₁₈y + a₁₉. Substituting nine input points creates equations for its nine coefficients; repeating the process for yᵗ creates an 18-equation system. section 8.1.1

How Geometry for Programmers covers it

How deep it goes
Defines the nine-term form and coefficient system, but stops short of detailing the complete symbolic solution.
Examples
Uses a transformed standard square, nine corresponding points, coordinate formulas, and an 18-equation implementation system.
What you can do after
Build it from scratch by substituting guiding points, solving for coefficients, and constructing paired coordinate polynomials.
How it connects
Connects biquadratic polynomials with polynomial transformations, bilinear transformations, polynomial coefficients, and symbolic equation systems.
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Related concepts

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.