路径或曲线中的几何

Basic Analytic Geometry & Curves

One could teach an entire school year's worth of analytic geometry. This will only cover some of the basics.

When you think of analytic geometry, you will most likely think of coordinate geometry. Analytic geometry is essentially using coordinates within geometry. The Cartesian Plane is normally applied in analytic geometry. The Cartesian Plane is the plane of all points with an $x$ and $y$ coordinate. For example, the point $(3,4)$ is $3$ units to the positive side, and $4$ units vertically.

Within analytic geometry, there are curves. Curves are normally defined as a subset of points within the analytic plane.

• A line is any curve where both $x$ and $y$ are to the first power in the equation of the curve. For example, $y=x$ is a line because $y$ and $x$ are both to the first power.
• A circle is an equation in the form $x^2+y^2=r^2$. Quadratics are equations where either $x$ or $y$ is raised to the second power.

Distance and angles have the same definitions as in Euclidean Geometry, but the formula for distance is given as

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

The formula for finding an angle between a line and a horizontal line is given as

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

where $m$ is the slope of the line.

There are multiple ways to "transform" a curve.

To list a few:

1: Changing $x$ to $(x-h)$ moves the curve to the right $h$.

2: Changing $y$ to $(y-k)$ moves the curve up $k$ units.

3: Dividing $x$ or $y$ will dilate the curve, either horizontally or vertically.

4: Changing $x$ to $x\cos(\theta) + y\sin(\theta)$ or $y$ to $-x\sin(\theta)+y\cos(\theta)$ rotates the curve by the angle $\theta$.

First off, a path is a line that isn't necessarily straight. It could be a curved line.

This line AB is two paths: A to B or B to A.

The difference between the two is that the idea of the straight line path is Euclidean Geometry, which is generally more straightforward and common. It helps to define the distance between two pints which is generally assumed to be the straight line path. There is only one straight line path between two points, but there is an infinite number of curved paths between two points.