1. Geometric properties: A circle is a set of all points on a plane equidistant from a given point. This given point is called the center of the circle, and the equidistant point is called the radius. A circle has many unique geometric properties, such as dividing it into two equal semicircles with a circumferential angle equal to a right angle. These properties make the circle play an important role in geometry.
2. Equation representation: A circle can be represented by an algebraic equation. The equation of a circle with the center as the origin and radius r is x 2+y 2 = r 2. This equation describes the relationship between the coordinates of all points on a circle. By solving this equation, we can find any point on the circle.
3. Parametric equation: In addition to polar coordinate system and rectangular coordinate system, a circle can also be expressed by parametric equation. The parameter equation of a circle with the center of the circle as the origin and radius r is x=r*cos(t) and y=r*sin(t), where t is the parameter. This representation makes it easier for us to describe the dynamic changes of points on a circle.
4. Tangent chord: The intersection line and tangent chord of a circle with other geometric figures are also important topics in the study of circles. For example, if a circle intersects a straight line, we can find the intersection point; When a circle is tangent to a straight line, we can solve the tangent equation. For the chord formed by the intersection of a circle and a straight line, the chord length can be calculated.
5. Application: The circle has many applications in real life, such as planetary orbits in astronomy and force fields in physics. These practical applications make the study of circles of great value.
In a word, the mathematical essence of the circle can be understood from geometric properties, equation representation, parametric equation, chord cutting, application and so on. Through the study of these aspects, we can better grasp the nature and application of the circle.