# 解析几何

1637年，笛卡兒在《方法论》的附录“几何”中提出了解析几何的基本方法。 以哲学观点写成的这部法语著作为后来牛顿莱布尼茨各自提出微积分学提供了基础。

## 基本理论

### 距离和角度

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},\!$

$\theta = \arctan(m)\!$

### 变化

a) y = f(x) = |x|       b) y = f(x+2)       c) y = f(x)-3       d) y = 1/2 f(x)

$x^2+y^2-1=0$

• $x$ 变为 $x-h$ ，使得图像向右移动 $h$ 个单位。
• $y$ 变为 $y-k$，使得图像向上移动 $k$ 个单位。
• $x$ 变为 $x/b$，使得图像以 $b$值拉伸。 (想象一下$x$ 被膨胀了)
• $y$ 变为 $y/a$，使得图像垂直拉伸。
• $x$ 变为 $x\cos A+ y\sin A$，将 $y$ 变为 $-x\sin A + y\cos A$，使得图像旋转 $A$ 个角度。

### 交集

$P$$Q$ 的交集可以通过同时解方程来求得：

$x^2+y^2 = 1$

$(x-1)^2+y^2 = 1$

$x^2+y^2 = 1$

$y^2=1-x^2$ 我们将$y^2$值带入其它等式当中：

$(x-1)^2+(1-x^2)=1$ 然后解 $x$:

$x^2 -2x +1 +1 -x^2 =1$

$-2x = -1$

$x=1/2$

$(1/2)^2+y^2 = 1$

$y^2 =3/4$

$y = \frac{\pm \sqrt{3}}{2}$

$\left(1/2,\frac{+ \sqrt{3}}{2}\right) \;\; \mathrm{and} \;\; \left(1/2,\frac{-\sqrt{3}}{2}\right)$

$y^2 = 3/4$

$y = \frac{\pm \sqrt{3}}{2}$

$\left(1/2,\frac{+ \sqrt{3}}{2}\right) \;\; \mathrm{and} \;\; \left(1/2,\frac{-\sqrt{3}}{2}\right)$

## 主题

$Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0$. 如果$Bxy$被考虑进去的话，就会常常用到旋转。这些问题常涉及到线性代数

## 例子

$F\left(\frac{a}{2},0\right)$, $G\left(\frac{a+b}{2},\frac{e}{2}\right)$, $H\left(\frac{b+c}{2},\frac{e+f}{2}\right)$, $I\left(\frac{c+d}{2},\frac{f+g}{2}\right)$, $X\left(\frac{a+b+c}{4},\frac{e+f}{4}\right)$, and $Y\left(\frac{a+b+c+d}{4},\frac{e+f+g}{4}\right).$

$AE=\sqrt{d^2+g^2}$

$XY=\sqrt{\frac{d^2}{16}+\frac{g^2}{16}}=\frac{\sqrt{d^2+g^2}}{4}.$

$AE\equiv 0\pmod{4}$

(见同余) 因此 $AE=4$.

## 注释

1. ^ Boyer, Carl B.. The Age of Plato and Aristotle. A History of Mathematics Second Edition. John Wiley & Sons, Inc. 1991: 94–95. ISBN 0-471-54397-7. "Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a strong resemblance to the use of coordinates, as illustrated above, it has sometimes been maintained that Menaechmus had analytic geometry. Such a judgment is warranted only in part, for certainly Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. It was shortcomings in algebraic notations that, more than anything else, operated against the Greek achievement of a full-fledged coordinate geometry."
2. ^ Boyer, Carl B.. Apollonius of Perga. A History of Mathematics Second Edition. John Wiley & Sons, Inc. 1991: 142. ISBN 0-471-54397-7. "The Apollonian treatise On Determinate Section dealt with what might be called an analytic geometry of one dimension. It considered the following general problem, using the typical Greek algebraic analysis in geometric form: Given four points A, B, C, D on a straight line, determine a fifth point P on it such that the rectangle on AP and CP is in a given ratio to the rectangle on BP and DP. Here, too, the problem reduces easily to the solution of a quadratic; and, as in other cases, Apollonius treated the question exhaustively, including the limits of possibility and the number of solutions."
3. ^ Boyer, Carl B.. Apollonius of Perga. A History of Mathematics Second Edition. John Wiley & Sons, Inc. 1991: 156. ISBN 0-471-54397-7. "The method of Apollonius in the Conics in many respects are so similar to the modern approach that his work sometimes is judged to be an analytic geometry anticipating that of Descartes by 1800 years. The application of references lines in general, and of a diameter and a tangent at its extremity in particular, is, of course, not essentially different from the use of a coordinate frame, whether rectangular or, more generally, oblique. Distances measured along the diameter from the point of tangency are the abscissas, and segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. The Apollonian relationship between these abscissas and the corresponding ordinates are nothing more nor less than rhetorical forms of the equations of the curves. However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed a posteriori upon a given curve in order to study its properties. There appear to be no cases in ancient geometry in which a coordinate frame of reference was laid down a priori for purposes of graphical representation of an equation or relationship, whether symbolically or rhetorically expressed. Of Greek geometry we may say that equations are determined by curves, but not that curves are determined by equations. Coordinates, variables, and equations were subsidiary notions derived from a specific geometric situation; [...] That Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry, was probably the result of a poverty of curves rather than of thought. General methods are not necessary when problems concern always one of a limited number of particular cases."
4. ^ 4.0 4.1 4.2 Boyer. The Arabic Hegemony. 1991: 241–242. "Omar Khayyam (ca. 1050–1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.""
5. ^ Glen M. Cooper (2003). "Omar Khayyam, the Mathematician", The Journal of the American Oriental Society 123.
6. ^ Stillwell, John. Analytic Geometry. Mathematics and its History Second Edition. Springer Science + Business Media Inc. 2004: 105. ISBN 0-387-95336-1. "the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments."
7. ^ Cooke, Roger. The Calculus. The History of Mathematics: A Brief Course. Wiley-Interscience. 1997: 326. ISBN 0-471-18082-3. "The person who is popularly credited with being the discoverer of analytic geometry was the philosopher René Descartes (1596–1650), one of the most influential thinkers of the modern era."
8. ^ 8.0 8.1 Katz 1998，pg. 442
9. ^ Katz 1998，pg. 436

## 引述

### 著作

• Katz, Victor J., A History of Mathematics: An Introduction (2nd Ed.), Reading: Addison Wesley Longman, 1998, ISBN 0-321-01618-1
• Boyer, Carl B., History of Analytic Geometry, Dover Publications, ISBN 978-0486438320
• Cajori, Florian, A History of Mathematics, AMS, ISBN 978-0821821022
• Struik, D. J., A Source Book in Mathematics, 1200-1800, Harvard University Press, ISBN 978-0674823556

### 文献

• Boyer, Carl B. Analytic Geometry: The Discovery of Fermat and Descartes, Mathematics Teacher 37, no. 3 (1944): 99-105
• Boyer, Carl B., Johann Hudde and space coordinates
• Bissell, C. C., Cartesian geometry: The Dutch contribution
• Pecl, J., Newton and analytic geometry
• Coolidge, J. L., The Beginnings of Analytic Geometry in Three Dimensions