9.7
双曲线几何
双曲线由所有到两个定点（称为焦点）的距离差的绝对值恒定的点组成。其标准方程为：
每个分支无限延伸，并逐渐趋近于两条渐近线，渐近线决定了曲线的趋近方向。参数 a 和 b 确定了双曲线的主要特征：a 表示从中心到横轴上各顶点的距离，而 b 则决定渐近线的斜率。渐近线的方程为。
以原点为中心、边长分别为 2a…
双曲线由两条开口的曲线组成，称为分支。如果 P 是曲线上的一点，则测量 P 到两个焦点的距离，这两个距离的绝对差值为常数。无论 P 在任一分支上的位置如何选择，该差值始终保持不变。
以原点为中心且沿 x 轴方向开口的双曲线的标准方程为 x²/a² - y²/b² = 1。
双曲线的每一分支均趋近于两条对角线，称为渐近线，这些渐近线引导曲线延伸至无穷远。
2a 表示沿横轴方向两个顶点之间的距离，而 2b 定义共轭轴的长度。
渐近线的方程取决于 a 和 b，这两个参数决定了中心矩形对角线的斜率。利用点斜式公式，可以写出渐近线的方程。
双曲线也出现在光学仪器中。在天文学中，卡塞格林望远镜使用一个抛物面主镜和一个双曲面副镜。主镜将入射的平行光线聚焦，而副镜与主镜共用一个焦点，将光线反射至其第二个焦点，并通过主镜上的孔洞形成图像。
View the full transcript and gain access to JoVE Core videos
Q1: What defines a hyperbola and how do its two branches relate to the foci?
A hyperbola consists of two open curves called branches, defined by a constant property: for any point P on either branch, the absolute difference of distances from P to two fixed points called foci remains constant. This defining relationship holds regardless of which branch or location on the curve you choose, making it a fundamental characteristic of hyperbolic geometry.
Q2: What is the standard equation of a hyperbola centered at the origin?
The standard equation for a hyperbola centered at the origin and opening along the x-axis is x² / a² − y² / b² = 1. Here, a represents the distance from the center to each vertex along the transverse axis, while b influences the shape and asymptote slopes. This form enables direct analysis and plotting of hyperbolic curves.
Q3: How do asymptotes guide a hyperbola, and what determines their slopes?
Each branch of a hyperbola approaches two diagonal lines called asymptotes that guide the curve toward infinity without ever touching them. The slopes of these asymptotes depend on both parameters a and b, which define the dimensions of a central rectangle. The diagonals of this rectangle, with dimensions 2a by 2b, directly determine the asymptote equations.
Q4: What does eccentricity measure in a hyperbola?
Eccentricity quantifies how open or spread a hyperbola is, always exceeding one for hyperbolas. It is calculated using the relationship between c (distance to foci) and a (distance to vertices). Understanding eccentricity helps distinguish hyperbolas from other conic sections like those studied in eccentricity of an ellipse, which have eccentricity less than one.
Q5: How are hyperbolic mirrors used in optical instruments like telescopes?
In a Cassegrain telescope, a hyperbolic secondary mirror works with a parabolic primary mirror to focus light precisely. The primary mirror focuses incoming parallel rays, and the hyperbolic secondary, sharing one focus with the primary, reflects them toward its second focus through a hole in the primary to form a clear image.
Q6: What is the relationship between the transverse and conjugate axes in a hyperbola?
The transverse axis defines the distance between the two vertices, measured as 2a, while the conjugate axis has length 2b. These perpendicular axes intersect at the center and determine the hyperbola's orientation and shape. Together, they form a central rectangle whose diagonals become the asymptotes guiding each branch.
Q7: Where are the foci located on a hyperbola, and how do they relate to the vertices?
For a hyperbola centered at the origin with horizontal transverse axis, the foci are located at (±c, 0), where c is calculated from the relationship c² = a² + b². The foci lie beyond the vertices on the transverse axis, and their separation determines the constant difference in distances that defines every point on the hyperbola.