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A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone (a cone with two nappes).It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice.
Menaechmus (Greek: Μέναιχμος, c. 380 – c. 320 BC) was an ancient Greek mathematician, geometer and philosopher [1] born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, who was known for his friendship with the renowned philosopher Plato and for his apparent discovery of conic sections and his solution to the then-long-standing problem of doubling the cube using the ...
Steiner claimed that the number of conics tangent to 5 given conics in general position is 7776 = 6 5, but later realized this was wrong. [2] The correct number 3264 was found in about 1859 by Ernest de Jonquières who did not publish because of Steiner's reputation, and by Chasles using his theory of characteristics, [3] and by Berner in 1865.
Another classic problem in enumerative geometry, of similar vintage to conics, is the Problem of Apollonius: a circle that is tangent to three circles in general determines eight circles, as each of these is a quadratic condition and 2 3 = 8. As a question in real geometry, a full analysis involves many special cases, and the actual number of ...
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form A x 2 + B x y + C y 2 + D x + E y + F = 0 with A , B , C not all zero. {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0{\text{ with ...
The problem comprises drawing lines from two points, meeting at a third point on the circumference of a circle and making equal angles with the normal at that point (specular reflection). Thus, its main application in optics is to solve the problem, "Find the point on a spherical convex mirror at which a ray of light coming from a given point ...
In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis , vertices , tangents and the pole and polar relationship between points and lines of the plane determined by the conic.
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: The eccentricity of a circle is 0. The eccentricity of an ellipse which is not a circle is between 0 and 1.