The pole theorem of circle is an important basic theorem in analytic geometry, which is widely used to calculate the parameter equation of circle and the distance from point to circle.
The pole of a circle means that for a given circle and a straight line in a plane rectangular coordinate system, every point on the straight line can be determined by a straight line passing through two points on the circle. Polar line refers to the set of all straight lines and one straight line tangent to a given circle in a plane rectangular coordinate system.
The theorem of poles and epipolar lines of a circle is as follows: for a given circle, if a point is selected as the pole in the plane rectangular coordinate system, the connecting line between the point and any point in the circle is the epipolar line of the point; On the contrary, for a straight line, if the set of all straight lines tangent to the straight line passing through the circle, that is, the figure composed of all polar lines is a point, then this point is the pole of the circle.
To understand this theorem, we must master the meaning of the word "pole". Even if the line connecting this point and another point on the circle passes through the line where this point is located, it is also called the pole of the circle on this line, and the line segment corresponding to this point and the circle on this line is the polar line. So the pole of a circle has a unique polar line, while a straight line has only one pole.
For example, in analytic geometry, we can transform the transformed circle into a standard equation and calculate the parameter equation of the circle by using the pole-pole theorem of the circle. At the same time, we can also solve the intersection position of a given circle and a straight line by pole-pole theorem, or find out where the farthest point from the circle is.
In a word, the pole theorem of a circle is one of the basic theorems in analytic geometry, which provides convenience for the calculation between a circle and a straight line. If we deeply understand the mathematical principle it expresses, we can better use it to complete geometric calculation and solve specific problems.
The geometric properties of polar lines are as follows:
1, any point on the projective plane has and only has one polar line for a fixed conic C, whereas any straight line on the projective plane has and only has one polar line for a fixed conic C, which can be directly deduced from the definition.
2. (Polar Matching Principle) For the same quadratic curve C, if the polar line of point P passes through point Q, then the polar line of point Q passes through point P. Conversely, if the pole of straight line P is on straight line Q, then the pole of straight line Q is on straight line P..
3. The pole of the connecting line between two points is the intersection of these two polar lines; The polar line of the intersection of two straight lines is the connecting line of the poles of these two straight lines. If there are two points A and B whose polar lines intersect with C, then according to the polar coordinate principle, C is on the polar line of A? A is on the polar line of C. Similarly, B is on the polar line of C. A straight line determined by two points indicates that AB is the polar line of C, that is, C is the pole of AB. Similar to the latter.