Elliptical Radial Gradient Generation Using Affine Transformation
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Solution Overview
Problem
Existing methods for drawing elliptical radial gradients are computationally intensive due to the complexity of the elliptical shape's equation, increasing the load on devices and making the process inefficient.
Innovation Solution
An image generating device and method that transforms the elliptical shape into a perfect circular shape using a transformation matrix, allowing for the calculation of gradation parameters based on the circular shape, which reduces the computational load by simplifying the calculation of color changing points and enables efficient drawing of the radial gradation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Shape
If the color changing points are calculated based on the locus of an elliptical shape, then an elliptical radial gradation can be drawn, but extremely complicated calculations become necessary and the device load becomes extremely heavy
Solution Approach 1:
The patent creates a simplified copy of the elliptical gradation problem by transforming it into a circular gradation problem. By applying an affine transformation that converts the elliptical coordinate system into a circular one, the device can calculate color changing points using simple circular geometry instead of complex elliptical equations, thereby reducing computational complexity while preserving the visual appearance of the elliptical gradation
Solution Approach 2:
The patent changes the parameter representation by introducing transformed coordinates (u, v) that map the elliptical space to a circular space. This parameter transformation allows the use of simple circular equations (u² + v² = r²) instead of complex elliptical equations, significantly reducing the computational burden while maintaining the ability to generate accurate elliptical radial gradations
2Shape
If the color changing points are calculated based on the locus of an elliptical shape, then an elliptical radial gradation can be drawn, but the load on the device becomes extremely heavy
Solution Approach 1:
The patent creates a simplified copy of the elliptical gradation problem by transforming it into a circular gradation problem. By applying an affine transformation that converts the elliptical coordinate system into a circular one, the device can calculate color changing points using simple circular geometry instead of complex elliptical equations, thereby reducing computational complexity while preserving the visual appearance of the elliptical gradation
Solution Approach 2:
The patent replaces the computationally intensive mathematical operations required for elliptical calculations with simpler operations in the transformed circular coordinate system. This substitution reduces the processing power and energy required, as the device performs basic arithmetic operations on transformed coordinates rather than solving complex elliptical equations
3Manufacturing precision
If multiple areas slightly differing in color value are drawn successively while gradually shifting the drawing position, then a gradation image can be generated, but complicated calculations are required for each area determination
Solution Approach 1:
The patent creates a simplified copy of the elliptical gradation problem by transforming it into a circular gradation problem. By applying an affine transformation that converts the elliptical coordinate system into a circular one, the device can calculate color changing points using simple circular geometry instead of complex elliptical equations, thereby reducing computational complexity while preserving the visual appearance of the elliptical gradation
Solution Approach 2:
The patent changes the parameter representation by introducing transformed coordinates (u, v) that map the elliptical space to a circular space. This parameter transformation allows the use of simple circular equations (u² + v² = r²) instead of complex elliptical equations, significantly reducing the computational burden while maintaining the ability to generate accurate elliptical radial gradations
Data Source
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AI summary
An image generating method comprises the steps of extracting elliptical shape parameters (specifying an elliptical shape), gradation pattern parameters (specifying a gradation pattern to be drawn in the elliptical shape) and drawing area parameters (specifying a drawing area) from an instruction for drawing an elliptical radial gradation, generating a transformation matrix for transforming the elliptical shape into a perfect circular shape based on the elliptical shape parameters, calculating an inverse matrix of the transformation matrix, transforming the gradation pattern parameters using the transformation matrix, transforming a first area containing the drawing area using the transformation matrix, drawing a radial gradation in the perfect circular shape in a second area containing the transformed first area based on the transformed gradation pattern parameters, and generating the elliptical radial gradation for the drawing area by inversely transforming the second area (in which the radial gradation has been drawn) using the inverse matrix.