Structural characteristics of geometry in high school mathematics space
Structural characteristics of 1. polyhedron
Two faces of a (1) prism are parallel to each other, the other faces are parallelograms, and the common edges of every two adjacent quadrangles are parallel.
Regular prism: a prism whose side is perpendicular to the bottom is called a regular prism, and a regular prism whose bottom is a regular polygon is called a regular prism. On the contrary, a regular prism has a regular bottom surface and rectangular side surfaces, and the side edges of the bottom surface are perpendicular to the bottom surface.
(2) The base of the pyramid is an arbitrary polygon, and the sides are triangles with common vertices.
Regular pyramid: A pyramid whose bottom is a regular polygon and whose vertices are projected on the bottom is called a regular pyramid. In particular, an equilateral regular triangular pyramid is called a regular tetrahedron. On the contrary, the base of a regular pyramid is a regular polygon, and the projection of its vertex on the base is the center of the regular polygon.
(3) A frustum can be obtained from a plane truncated pyramid parallel to the bottom surface, and its upper and lower bottom surfaces are similar polygons.
2. Structural characteristics of the rotating body
(1) A cylinder can be obtained by rotating a rectangle around a straight line with one side.
(2) Turn the right triangle around a straight line with right angles to get a cone.
(3) The frustum can be a right-angled trapezoid rotating once around the straight line where the right-angled waist is located, or an isosceles trapezoid rotating half a circle around the straight line where the centers of the upper and lower bottom surfaces are located, or a plane truncated cone parallel to the bottom surface.
(4) The ball can be a semicircle that rotates once around the diameter, or it can be a semicircle that rotates once around the diameter.
3. Three views of space geometry
Three views of space geometry are obtained by parallel projection. Under this projection, the shadow left by the plane figure parallel to the projection plane is congruent with the plane figure, and its shape and size are equal. Three views include front view, side view and top view.
Three-view length characteristics:? Long alignment, equal width, high level? That is, the front view is as high as the side view, the front view is as long as the top view, and the side view is as wide as the top view. If the surfaces of two adjacent objects intersect, the intersection line of the surfaces is their dividing line. Attention should be paid to the drawing of virtual and real lines in three views.
4. Intuition of space geometry
Orthographic drawings of space geometry are often drawn by oblique survey, and the basic steps are as follows:
(1) Draw the bottom surface of the geometry.
Take the X-axis and Y-axis perpendicular to each other in the known graph, and the two axes intersect at the O-point. When drawing a straight view, draw it as the corresponding X? Axis, y? Axis, the two axes intersect at point o? , and make? x？ o？ y？ =45? Or 135? It is known that the line segments parallel to the X and Y axes in the graph are parallel to X? Axis, y? Axis. It is known that in orthographic drawing, the length of the line segment parallel to the X-axis is unchanged, and the length of the line segment parallel to the Y-axis is half of the original length.
(2) the height of the drawing geometry
In a known graph, the Z axis passing through the O point is perpendicular to the xOy plane, and the corresponding Z? Axis, also perpendicular to x? o？ y？ Plane, the line segment parallel to the Z axis in a known figure, is still parallel to Z under direct vision? The axis and length are the same.
Requirements of high altitude geometric structure test site
The expanded diagram and three views of 1. geometry are still hot spots in the college entrance examination.
2. The proposition of combining three views with other knowledge points is the trend of new textbooks to examine students' three views and geometric quantity calculation.
3. Focus on the topic of spatial geometric structure characteristics with three views as the proposition background.