1, the definition of polar diameter:
In plane geometry, consider a circle whose center is O and radius is R. The arc length AB between any two points A and B on the circle is the arc length corresponding to the central angle θ. Polar diameter refers to the maximum arc length AB between two points A and B on a circle. In other words, the polar diameter is the longest chord from the center o to a point on the circle.
2, the nature of the polar diameter:
Polar diameter is the diameter of a circle: according to the definition of polar diameter, polar diameter is the maximum arc length between two points on a circle, and the diameter of a circle is the maximum arc length between two points on a circle, so polar diameter is the diameter of a circle.
The polar diameter is twice the diameter: let d be the diameter and r be the radius of the circle, then the length of the polar diameter is 2r, that is, the polar diameter is equal to twice the diameter.
Relationship between polar diameter and central angle: There is a special relationship between polar diameter and its corresponding central angle. For any point p on a circle, the line segment OP between it and the center o is the polar diameter. Let the length of the polar diameter be r and the central angle be θ, then according to the properties of the circle, the relationship between the polar diameter and the central angle is satisfied: arc length AB = rθ.
3, the application of polar diameter:
The concept of polar diameter is widely used in mathematics and geometry, especially in solving problems related to circles. The following are some application examples:
Parametric equation of circle: For the intersection of straight line and circle, the parametric equation of circle can be constructed by the concept of polar diameter. Let the radius of the circle be R, the center of the circle be O, the intersection point of the straight line and the circle be P, and the distance between the straight line and the center be D, then a parametric equation can be established by using the properties of the polar diameter, such as (x-d) 2+y 2 = r 2.
Polar coordinate system: Polar diameter is one of the important elements of polar coordinate system. In polar coordinate system, the position of a point is determined by the polar diameter and the polar angle. The polar diameter represents the distance between a point and a pole, and the polar angle represents the angle of the point relative to the polar axis.