For a homogeneous polynomial of degree , its first polar with respect to is the equation , of degree when this polynomial is nonzero. Every singular point of lies on every such polar. Polars turn the Bézout's theorem intersection count into bounds on singularities and tangencies. The equation is covariant under a linear change of homogeneous coordinates, by the chain rule.
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A polar curve is a graph that represents a relationship between a point in the polar coordinate system defined by its distance from a reference point (the pole) and its angle from a reference direction (usually the positive x-axis). In polar coordinates, a point is represented as \((r, \theta)\), where: - \(r\) is the radial distance from the origin (the pole) to the point.