1, definition: hyperbola is a special curve on a plane, and its definition is that the difference between the distances to two given points is equal to a constant. These two given points are called focus and the constant is called eccentricity.

2. Focus and diameter of hyperbola: For hyperbola, focus is an important element in its definition. The straight line passing through the focal point is called the diameter of hyperbola, and the difference between the distance and the diameter of each point on hyperbola is constant.

3. Asymptote: The hyperbola has two asymptotes, which tend to be parallel to the branches of the curve at infinity. The slope of the asymptote is equal to the eccentricity.

4. Radius of curvature: Every point on the hyperbola has a radius of curvature, and the radius of curvature is inversely proportional to the distance from the point to the focus. This is an important aspect of geometric properties of hyperbola.

5. Symmetry: A hyperbola is a figure that is symmetrical about two coordinates. This means that the properties of hyperbola on the coordinate axis are consistent with those on the curve.

6. Area: The area between two branches of hyperbola is infinite. The area calculation of hyperbola needs integration, which is a unique feature of its geometric properties.

7. The property of equal area: Although the area between the two branches of hyperbola is infinite, in a specific range, the areas of the two branches of hyperbola are equal. This is one of the applications of hyperbola in mathematics and physics.

8. Right-angled triangle with focus: If the distance between a point on the hyperbola and the two focuses is A and B, then the hypotenuse length of the right-angled triangle with this point as the vertex is C, and A? +b？ =c？ . This is an important triangular property of hyperbola.

9. Chord length from the focus to a point on the curve: the chord length from the focus to a point on the hyperbola is proportional to the difference between the distances from that point and the two focuses. This is an interesting geometric property of hyperbola in chord length.

Through the in-depth study of these typical examples, we can better understand the geometric properties of hyperbola and apply it to mathematical and physical problems more flexibly.