Describe the underlying math equations for a collision between a ray and a sphere.

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Ray tracing is a technique that has been around for quite a long time. It allows for images to be generated given some mathematical description of objects in the scene, eye placement and viewport definition. As rays are cast through pixels in the image area, math calculations are used to determine which objects are collided with and the resulting color values associated at each pixel location.
A nice walk through on the underlying technique of ray tracing is provided by:
https://www.scratchapixel.com/lessons/3d-basic-rendering/minimal-ray-tracer-rendering-simple-shapes
In this assignment you should:
1. Summarize the ray tracing technique.
2. Describe the underlying math equations for a collision between a ray and a sphere.
3. Describe how color values at a pixel are calculated.
4. Describe how the scene is setup.
5. Describe how shadows, reflection, and refraction work in these simulated scenes.
6. Additional discussion on the state of the art in ray tracing. Why has it taken so long for ray tracing to take off? What recent innovations has allowed for the expanded use of ray tracing technology? Is real-time ray tracing a thing?
7. Provide a ray traced image and describe why this image is better or not than a raster-ized image.
You should submit a report (as a Word document) that is as detailed as possible. This is a topic that you are expected to delve into. Additional resources should be used and cited. Submit your detailed report by the due date posted on Blackboard. No late assignments will be accepted.

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