How does a conic create a dual relationship between an external point and a line?
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, a conic section creates a dual relationship between an external point and a line through the concept of polarity.1
For a given conic, an external point is associated with a unique line, known as the polar of that point. This line is defined by joining the two points of tangency where the tangents from meet the conic. Conversely, the point is referred to as the pole of that line.12
This relationship is a fundamental application of the principle of duality, which allows for the systematic interchange of the terms "point" and "line" within geometric theorems. By associating each point with a line and each line with a point, mathematicians can derive new geometric results from existing ones, effectively doubling the number of theorems available in the field. While this specific construction using tangents applies to points outside the conic, the concept of poles and polars can be extended to other positions, maintaining the reciprocal nature of the dual relationship.3145