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How do homogeneous coordinates represent ordinary points and points at infinity?

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Projective drawingThe sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady. — projective geometry Projective drawing
The sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.

In projective geometry, homogeneous coordinates provide a uniform way to represent both ordinary points and points at infinity using triples of numbers.1

Representation of Points

Introduced by Julius Plücker, homogeneous coordinates represent a point in the plane as a triple $(x, y, z)$ .1

Ordinary Points: An ordinary point in the Euclidean plane with Cartesian coordinates $(a, b)$ is represented in homogeneous coordinates by the triple $(a, b, 1)$ . More generally, any triple $(k a, k b, k)$ for a non-zero constant $k$ represents the same point, reflecting the "homogeneous" nature of the system where only the ratios of the coordinates matter.1
Points at Infinity: Points at infinity are represented by triples where the third coordinate is zero, typically in the form $(x, y, 0)$ . Because these points do not have a finite location in the Euclidean plane, they are used to represent the direction in which parallel lines appear to converge, effectively allowing the projective plane to treat these points as standard elements of the geometry.21