“quadric”的概念、定义、翻译、参考文献-科学参考

quadric

Mathematics

A locus in 3-dimensional space that can be represented in a Cartesian coordinate system by a polynomial equation in x, y, and z of degree two; that is, an equation of the form

$a x^{2} + b y^{2} + c z^{2} + 2 f y z + 2 g z x + 2 h x y + 2 u x + 2 v y + 2 w z + d = 0,$

where the constants a, b, c, f, g, and h are not all zero. When the equation represents a non-empty locus, it can be reduced by translation and rotation to one of the following canonical forms:
1. (i) Ellipsoid: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1.$
2. (ii) Hyperboloid of one sheet: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1.$
3. (iii) Hyperboloid of two sheets: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = - 1.$
4. (iv) Elliptic paraboloid: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{2 z}{c} .$
5. (v) Hyperbolic paraboloid: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = \frac{2 z}{c} .$
6. (vi) Quadric cone: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{z^{2}}{c^{2}} .$
7. (vii) Elliptic cylinder: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1.$
8. (viii) Hyperbolic cylinder: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.$
9. (ix) Parabolic cylinder: $\frac{x^{2}}{a^{2}} = \frac{2 y}{b} .$
10. (x) Pair of non-parallel planes: $\frac{x^{2}}{a^{2}} = \frac{y^{2}}{b^{2}} (that is, y = \pm \frac{b}{a} x) .$
11. (xi) Pair of parallel planes: $\frac{x^{2}}{a^{2}} = 1 (that is, x = \pm a) .$
12. (xii) Plane: $\frac{x^{2}}{a^{2}} = 0 (that is, x = 0) .$
13. (xiii) Line: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 0 (that is, x = y = 0) .$
14. (xiv) Point: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 0 (that is, x = y = z = 0) .$
Forms (i)–(v) are the non-degenerate quadrics.

单词	quadric
释义	quadric Mathematics A locus in 3-dimensional space that can be represented in a Cartesian coordinate system by a polynomial equation in x, y, and z of degree two; that is, an equation of the form $a x^{2} + b y^{2} + c z^{2} + 2 f y z + 2 g z x + 2 h x y + 2 u x + 2 v y + 2 w z + d = 0,$ where the constants a, b, c, f, g, and h are not all zero. When the equation represents a non-empty locus, it can be reduced by translation and rotation to one of the following canonical forms: (i) Ellipsoid: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1.$ (ii) Hyperboloid of one sheet: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1.$ (iii) Hyperboloid of two sheets: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = - 1.$ (iv) Elliptic paraboloid: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{2 z}{c} .$ (v) Hyperbolic paraboloid: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = \frac{2 z}{c} .$ (vi) Quadric cone: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = \frac{z^{2}}{c^{2}} .$ (vii) Elliptic cylinder: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1.$ (viii) Hyperbolic cylinder: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1.$ (ix) Parabolic cylinder: $\frac{x^{2}}{a^{2}} = \frac{2 y}{b} .$ (x) Pair of non-parallel planes: $\frac{x^{2}}{a^{2}} = \frac{y^{2}}{b^{2}} (that is, y = \pm \frac{b}{a} x) .$ (xi) Pair of parallel planes: $\frac{x^{2}}{a^{2}} = 1 (that is, x = \pm a) .$ (xii) Plane: $\frac{x^{2}}{a^{2}} = 0 (that is, x = 0) .$ (xiii) Line: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 0 (that is, x = y = 0) .$ (xiv) Point: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 0 (that is, x = y = z = 0) .$ Forms (i)–(v) are the non-degenerate quadrics.
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