Matrix representation of conic sections (original) (raw)
In geometria, una sezione conica può essere rappresentata in forma matriciale, ossia attraverso l'impiego di matrici.
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| dbo:abstract | 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. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not parallel to the coordinate system. Conic sections (including degenerate ones) are the sets of points whose coordinates satisfy a second-degree polynomial equation in two variables, By an abuse of notation, this conic section will also be called Q when no confusion can arise. This equation can be written in matrix notation, in terms of a symmetric matrix to simplify some subsequent formulae, as The sum of the first three terms of this equation, namely is the quadratic form associated with the equation, and the matrix is called the matrix of the quadratic form. The trace and determinant of are both invariant with respect to rotation of axes and translation of the plane (movement of the origin). The quadratic equation can also be written as where is the homogeneous coordinate vector in three variables restricted so that the last variable is 1, i.e., and where is the matrix The matrix is called the matrix of the quadratic equation. Like that of , its determinant is invariant with respect to both rotation and translation. The 2 × 2 upper left submatrix (a matrix of order 2) of AQ, obtained by removing the third (last) row and third (last) column from AQ is the matrix of the quadratic form. The above notation A33 is used in this article to emphasize this relationship. (en) In geometria, una sezione conica può essere rappresentata in forma matriciale, ossia attraverso l'impiego di matrici. (it) Reductie van een kegelsnede is een draaiing en/of verschuiving van het assenkruis zodat de vergelijking van de kegelsnede tot een eenvoudige en herkenbare standaardvorm wordt herleid. Eerst wordt indien nodig een draaiing uitgevoerd, een bewerking die gebaseerd. Daarna volgt eventueel als tweede stap een verschuiving van het geroteerde assenkruis. Door dezelfde bewerkingen te gebruiken in drie dimensies kan op analoge manier een kwadriek gereduceerd worden naar een standaardvorm. (nl) |
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| rdfs:comment | In geometria, una sezione conica può essere rappresentata in forma matriciale, ossia attraverso l'impiego di matrici. (it) Reductie van een kegelsnede is een draaiing en/of verschuiving van het assenkruis zodat de vergelijking van de kegelsnede tot een eenvoudige en herkenbare standaardvorm wordt herleid. Eerst wordt indien nodig een draaiing uitgevoerd, een bewerking die gebaseerd. Daarna volgt eventueel als tweede stap een verschuiving van het geroteerde assenkruis. Door dezelfde bewerkingen te gebruiken in drie dimensies kan op analoge manier een kwadriek gereduceerd worden naar een standaardvorm. (nl) 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. The technique does not require putting the equation of a conic section into a standard form, thus making it easier to investigate those conic sections whose axes are not parallel to the coordinate system. The sum of the first three terms of this equation, namely (en) |
| rdfs:label | Rappresentazione matriciale delle coniche (it) Matrix representation of conic sections (en) Reductie (kegelsnede) (nl) |
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