In daily life, everything people see is three-dimensional. The so-called points, lines, surfaces, bodies and angles are all abstract concepts. This kind of abstraction not only abandons the essential elements of objects such as colors and materials, but also ignores the space occupied: points are not too big or too small, lines are not too wide or narrow, and surfaces are not too thick or thin. These abstract concepts are not realistic in themselves, but only conceptual.
In Euclidean geometric space, the linear distance between two points is essential, and the length of a line segment can be defined by the linear distance between two points. The so-called two lines are parallel, which means that the distance between them is equal everywhere; The so-called parallelogram is two groups of quadrangles with equal opposite sides; The so-called congruence between two graphs means that one graph is transformed from another graph, and the distance between any two points on the graph remains unchanged during this transformation; Pythagorean theorem describes the relationship between three sides of a right triangle; Trigonometric function describes the relationship between acute angle and side length in right triangle.
On the basis of line segment length, people define the measurement of area on the plane and volume in space, and these measurements are based on the straight line distance between two points.
Translation, rotation and axial symmetry are the most vivid parts in the content of "graphics and geometry" in primary school mathematics. Taking "the movement of graphics" as the topic, the movement needs a reference object, both translation and rotation reference objects are a ray, and the axial symmetry reference object is a straight line. This movement has a * * * feature, that is, the distance between two points remains unchanged after the movement, thus ensuring the shape of the object remains unchanged after the movement. This movement is called rigid body movement.
Question 23: How to understand length, area and volume? The three concepts of length, area and volume are all measures of graphics: length is the measure of one-dimensional space graphics; Area is a measure of two-dimensional space graphics; Volume is a measure of three-dimensional space graphics. These three measurements are based on the length of a straight line segment, which is based on the straight line distance between two points, that is, the measurement is based on the straight line distance between two points. To measure, it is necessary to determine the measurement unit. The so-called measurement is to calculate how many measurement units the graph to be measured contains. The measurement unit of area and volume is based on the length unit of one-dimensional space, which is artificially specified.
Question 24: How to understand translation, rotation and axial symmetry? Judging the motion of an object requires a reference object, translation: the reference object is a ray, and the distance between all points on the map and the ray is constant. The movement that moves the same distance along the ray direction is called translation. Rotation: the reference object is a ray, which means that the distance between all points on the graph and the origin of the ray is constant, and the movement that moves at the same angle relative to the ray is rotation. Axisymmetry: the reference object is a straight line, which means that the figure turns to the other side of the straight line, the distance between the corresponding point and the straight line is equal, and the movement of the connecting line perpendicular to the straight line is axisymmetric.