<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left right" columnspacing="0em 1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>g</mi><msup><mi>z</mi><mn>3</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mi>h</mi><msup><mi>z</mi><mn>2</mn></msup><mi>w</mi><mo>+</mo><mi>j</mi><mi>z</mi><msup><mi>w</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi><msup><mi>w</mi><mn>3</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mi>m</mi><msup><mi>z</mi><mn>2</mn></msup><mo>+</mo><mi>p</mi><mi>z</mi><mi>w</mi><mo>+</mo><mi>q</mi><msup><mi>w</mi><mn>2</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mi>r</mi><mi>z</mi><mo>+</mo><mi>s</mi><mi>w</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mi>t</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} gz^3 & + hz^2w + jzw^2 + kw^3 & \\ & + mz^2 + pzw + qw^2 & \\ & + rz + sw & \\ & + t & = 0 \end{aligned} </annotation></semantics></math>gz3+hz2w+jzw2+kw3+mz2+pzw+qw2+rz+sw+t=0
The plane is skewed, rotated, and translated by a linear transformation.
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>z</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>w</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>α</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>β</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>γ</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>δ</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>x</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>y</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>+</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>ζ</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>η</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>z</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>α</mi><mi>x</mi><mo>+</mo><mi>β</mi><mi>y</mi><mo>+</mo><mi>ζ</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>w</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>γ</mi><mi>x</mi><mo>+</mo><mi>δ</mi><mi>y</mi><mo>+</mo><mi>η</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex"> \begin{aligned} \begin{bmatrix} z \\ w \end{bmatrix} & = \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} +\begin{bmatrix} \zeta \\ \eta \end{bmatrix} \\ z & = \alpha x + \beta y + \zeta \\ w & = \gamma x + \delta y + \eta \end{aligned} </annotation></semantics></math>[zw]zw=[αγβδ][xy]+[ζη]=αx+βy+ζ=γx+δy+η
Substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mi>x</mi><mo>+</mo><mi>β</mi><mi>y</mi><mo>+</mo><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\alpha x + \beta y + \zeta</annotation></semantics></math>αx+βy+ζ and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi><mi>x</mi><mo>+</mo><mi>δ</mi><mi>y</mi><mo>+</mo><mi>η</mi></mrow><annotation encoding="application/x-tex">\gamma x + \delta y + \eta</annotation></semantics></math>γx+δy+η for z and w in the general equation, simplify by collecting like terms in respective powers of x and y, and solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>α, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>β, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math>γ, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math>δ, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ζ</mi></mrow><annotation encoding="application/x-tex">\zeta</annotation></semantics></math>ζ, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>η, a and b so that
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">y^2 = x^3 + ax + b . </annotation></semantics></math>y2=x3+ax+b.