FOV angle conversion
FOV angle conversion - horizontal to vertical
The following equation is related to the field of view (FOV) in computer graphics, particularly in 3D rendering. The field of view is the extent of the observable world that is seen at any given moment by the camera. This equation is used to convert the vertical field of view (FOVy) to the horizontal field of view (FOVx) based on the aspect ratio of the display.
\[ \text{fov}_x = 2 \cdot \arctan\left( \tan\left( \frac{\text{fov}_y}{2} \right) \cdot \text{aspect}_x \right) \]
Here’s a step-by-step explanation of how this equation works:
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Vertical Field of View ((\text{fov}_y)): This is the angle in degrees or radians that describes how much of the scene can be seen vertically. It is given or defined for the camera or perspective.
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Aspect Ratio ((\text{aspect}_x)): This is the ratio of the width to the height of the display or the viewport. It can be calculated as: \[ \text{aspect}_x = \frac{\text{width}}{\text{height}} \]
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Conversion to Tangent: The vertical field of view is halved and then converted to the tangent. The tangent of half the vertical field of view gives a ratio that represents the visible extent in the vertical direction at a distance of 1 unit away: \[ \tan\left( \frac{\text{fov}_y}{2} \right) \]
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Scaling by Aspect Ratio: This tangent value is then multiplied by the aspect ratio to scale it to the horizontal direction. Essentially, this adjusts the vertical field of view to the appropriate width based on the aspect ratio: \[ \tan\left( \frac{\text{fov}_y}{2} \right) \cdot \text{aspect}_x \]
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Conversion Back to Angle: The arctangent (inverse tangent) is then taken to convert this ratio back into an angle, representing the horizontal field of view in radians: \[ \arctan\left( \tan\left( \frac{\text{fov}_y}{2} \right) \cdot \text{aspect}_x \right) \]
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Doubling the Angle: Finally, the result is doubled because the initial tangent calculation was based on half of the field of view. This gives the full horizontal field of view: \[ \text{fov}_x = 2 \cdot \arctan\left( \tan\left( \frac{\text{fov}_y}{2} \right) \cdot \text{aspect}_x \right) \]
This equation is crucial in 3D graphics to ensure that the perspective projection maintains the correct proportions and visual consistency, especially when the aspect ratio changes.
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