Построить центральную проекцию вершин A1, B1, C1, D1 фигуры плоскости ABCD, если фигура ABCD параллельна плоскости проекции Π₁, S — центр проекции, а A₁ — центральная проекция вершины A. ---**Chart/Diagram Description:**
* **Type:** Three-dimensional geometric diagram.
* **Main Elements:**
* **Plane:** A plane labeled `Π₁` (Pi sub 1) is depicted as a parallelogram in the lower part of the diagram, representing a flat surface.
* **Points:**
* `S`: A point located in the upper-left region.
* `A`: A point located centrally, connected to S, D, B, and A₁.
* `D`: A point located above the plane, connected to A and C.
* `C`: A point located on the plane `Π₁`, connected to D and B.
* `B`: A point located on the plane `Π₁`, connected to C and A.
* `A₁`: A point located in the lower-right region, connected to A.
* **Lines:**
* Straight line segments connect the following pairs of points: `SA`, `AD`, `DC`, `CB`, `BA`, `AA₁`.
* **Shapes:**
* A quadrilateral `ABCD` is formed by the segments `AD`, `DC`, `CB`, and `BA`.
* The segment `CB` lies on the plane `Π₁`.
* **Relative Position and Direction:**
* Points `C` and `B` lie on the plane `Π₁`.
* Points `D` and `A` are positioned above the plane `Π₁`.
* Point `S` is positioned above point `A`.
* Point `A₁` is positioned below and to the right of point `A`.
* The quadrilateral `ABCD` appears to be a figure whose base `CB` rests on the plane `Π₁`, with the rest of the figure extending upwards from the plane.
* **Labels and Annotations:** The plane is labeled `Π₁`. The vertices are labeled `S`, `A`, `D`, `C`, `B`, and `A₁`.
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Central projection is a fundamental geometric method used to project points from three-dimensional space onto a two-dimensional plane. The process involves a projection center S, a projection plane Pi-1, and projection rays that connect the center to each point being projected. When a ray from S passes through a point P, it intersects the projection plane at point P-1, which is the central projection of P.
Now let's set up our specific problem. We have a quadrilateral ABCD that is parallel to the projection plane Pi-1. The projection center S is positioned above the plane. We are given that A-1 is the central projection of vertex A, shown by the ray from S through A intersecting the plane at A-1. Our task is to find the projections B-1, C-1, and D-1 of the remaining vertices. Since ABCD is parallel to the projection plane, the projected figure will maintain the same shape as the original quadrilateral.
Now we construct the projection rays systematically. First, we already have ray SA passing through the projection center S and vertex A, extending to intersect the projection plane at A-1. Next, we draw ray SB from center S through vertex B. Similarly, we construct ray SC from S through vertex C, and ray SD from S through vertex D. Each ray represents the path of projection from the center through each vertex. These rays will intersect the projection plane to give us the projected vertices. The key principle is that each projection ray is a straight line connecting the projection center to the original vertex and extending to meet the projection plane.