derive the bi-quadratic equations. start from prpjective equations. use the second taylor series. sopve it step by step. please if you are not sure don't solve it please do not provide solution in image format thank you! ### Bi-quadraticA bi-quadratic function can be described by the following set of equations:\[ x' = a_1 + a_2x + a_3y + a_4x^2 + a_5y^2 + a_6xy \]\[ y' = a_7 + a_8x + a_9y + a_{10}x^2 + a_{11}y^2 + a_{12}xy \]where:- \( x \) and \( y \) are the input variables.- \( x' \) and \( y' \) are the output variables.- \( a_1, a_2, ..., a_{12} \) are constant coefficients.This pair of equations represents a quadratic transformation in two variables. The quadratic terms \( x^2, y^2, \) and \( xy \) allow for more complex mappings than linear transformations, accommodating curvature in the relationships between variables. This type of transformation is useful in various fields such as computer graphics, physics, and engineering for applications that require modeling more intricate patterns and behaviors.There are no graphs or diagrams included in this image. ## Projective Transformation### FormulasA projective transformation can be represented by the following equations:\[ x' = \frac{a_1 x + a_2 y + b_1}{c_1 x + c_2 y + 1} \]\[ y' = \frac{a_3 x + a_4 y + b_1}{c_1 x + c_2 y + 1} \]### Explanation- \( x' \) and \( y' \) denote the transformed coordinates - \( x \) and \( y \) denote the original coordinates - \( a_1, a_2, a_3, a_4, b_1, c_1, c_2 \) are constants that dictate how the coordinates are transformed.In these formulas, both the numerator and the denominator are linear combinations of the original coordinates \( x \) and \( y \) and a constant term. This type of transformation can handle a wide variety of transformations including translation, scaling, rotation, and perspective transformations.### Detailed Graphical RepresentationIn a typical projective transformation, points in a plane are transformed in a way that lines remain lines, but parallel lines may not remain parallel and may instead converge to a point. This is commonly used in computer vision and graphics to simulate the perspective effect and in geographic coordinate transformations.To visualize, consider a grid of points in the original coordinate system. After applying the transformation formulas for \( x' \) and \( y' \), the grid of points distorts, illustrating the effect of the transformation parameters on the plane.

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derive the bi-quadratic equations. start from prpjective equations. use the second taylor series. sopve it step by step. please if you are not sure don't solve it

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

derive the bi-quadratic equations. start from prpjective equations. use the second taylor series. sopve it step by step. please if you are not sure don't solve it

please do not provide solution in image format thank you!

### Bi-quadratic

A bi-quadratic function can be described by the following set of equations:

\[ x' = a_1 + a_2x + a_3y + a_4x^2 + a_5y^2 + a_6xy \]
\[ y' = a_7 + a_8x + a_9y + a_{10}x^2 + a_{11}y^2 + a_{12}xy \]

where:
- \( x \) and \( y \) are the input variables.
- \( x' \) and \( y' \) are the output variables.
- \( a_1, a_2, ..., a_{12} \) are constant coefficients.

This pair of equations represents a quadratic transformation in two variables. The quadratic terms \( x^2, y^2, \) and \( xy \) allow for more complex mappings than linear transformations, accommodating curvature in the relationships between variables. This type of transformation is useful in various fields such as computer graphics, physics, and engineering for applications that require modeling more intricate patterns and behaviors.

There are no graphs or diagrams included in this image.

## Projective Transformation

### Formulas

A projective transformation can be represented by the following equations:

\[ x' = \frac{a_1 x + a_2 y + b_1}{c_1 x + c_2 y + 1} \]

\[ y' = \frac{a_3 x + a_4 y + b_1}{c_1 x + c_2 y + 1} \]

### Explanation

- \( x' \) and \( y' \) denote the transformed coordinates
- \( x \) and \( y \) denote the original coordinates
- \( a_1, a_2, a_3, a_4, b_1, c_1, c_2 \) are constants that dictate how the coordinates are transformed.

In these formulas, both the numerator and the denominator are linear combinations of the original coordinates \( x \) and \( y \) and a constant term. This type of transformation can handle a wide variety of transformations including translation, scaling, rotation, and perspective transformations.

### Detailed Graphical Representation

In a typical projective transformation, points in a plane are transformed in a way that lines remain lines, but parallel lines may not remain parallel and may instead converge to a point. This is commonly used in computer vision and graphics to simulate the perspective effect and in geographic coordinate transformations.

To visualize, consider a grid of points in the original coordinate system. After applying the transformation formulas for \( x' \) and \( y' \), the grid of points distorts, illustrating the effect of the transformation parameters on the plane.

Formula Formula A polynomial with degree 2 is called a quadratic polynomial. A quadratic equation can be simplified to the standard form: ax² + bx + c = 0 Where, a ≠ 0. A, b, c are coefficients. c is also called "constant". 'x' is the unknown quantity

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