The Steiner conic or more precisely Steiner's generation of a conic, named after the Swiss mathematician Jakob Steiner, is an alternative method to define a non-degenerate projective conic section in a projective plane over a field.

The usual definition of a conic in projective space uses a quadratic form. Another alternative definition of a conic uses a hyperbolic polarity. It is due to K. G. C. von Staudt and sometimes called a von Staudt conic. The disadvantage of von Staudt's definition is that it only works when the underlying field has odd characteristic.

A perspective mapping ${\displaystyle \pi }$ of a pencil ${\displaystyle B(U)}$ onto a pencil ${\displaystyle B(V)}$ is a bijection (1-1 correspondence) such that corresponding lines intersect on a fixed line ${\displaystyle a}$, which is called the axis of the perspectivity ${\displaystyle \pi }$ (figure 2).

A projective mapping is a finite product of perspective mappings.

Simple example: If one shifts in the first diagram point ${\displaystyle U}$ and its pencil of lines onto ${\displaystyle V}$ and rotates the shifted pencil around ${\displaystyle V}$ by a fixed angle ${\displaystyle \varphi }$ then the shift (translation) and the rotation generate a projective mapping ${\displaystyle \pi }$ of the pencil at point ${\displaystyle U}$ onto the pencil at ${\displaystyle V}$. From the inscribed angle theorem one gets: The intersection points of corresponding lines form a circle.

Examples of commonly used fields are the real numbers ${\displaystyle \mathbb {R} }$, the rational numbers ${\displaystyle \mathbb {Q} }$ or the complex numbers ${\displaystyle \mathbb {C} }$. The construction also works over finite fields, providing examples in finite projective planes.

Remark: The fundamental theorem for projective planes states, that a projective mapping in a projective plane over a field (pappian plane) is uniquely determined by prescribing the images of three lines. That means that, for the Steiner generation of a conic section, besides two points ${\displaystyle U,V}$ only the images of 3 lines have to be given. These 5 items (2 points, 3 lines) uniquely determine the conic section.

Remark: The notation "perspective" is due to the dual statement: The projection of the points on a line ${\displaystyle a}$ from a center ${\displaystyle Z}$ onto a line ${\displaystyle b}$ is called a perspectivity (see below).