Homogeneous Co-ordinates in Action?

📌 CONCEPT: In homogeneous coordinates, a point in 3D space is represented as (x : y : z), where the colon is used to separate the coordinates, and the point at infinity is represented by (x : y : 0).

📐 RULE / FORMULA: The equation of a sphere in homogeneous coordinates is given by x^2 + y^2 + z^2 = r^2, where r is the radius of the sphere.

💡 WORKED EXAMPLE: For a sphere with radius 5 units, centered at the origin, the equation in homogeneous coordinates is x^2 + y^2 + z^2 = 25. To find the point on the sphere closest to (4, 3, 0), we can use the equation to find the point on the sphere with the minimum distance. We can do this by finding the point on the sphere with the minimum value of the dot product of the position vectors of the two points.

⚠️ COMMON MISTAKE: Students often get the equation of the sphere in homogeneous coordinates wrong by not using the correct formula or not considering the point at infinity.