The key features of a central projection, particularly between parallel planes, involve how it transforms geometric elements. According to the provided reference, a central projection of a plane on a parallel plane uniquely maps between points and maps lines on lines and multiplies distances by a constant factor.
Understanding Central Projection
Central projection is a type of perspective projection where points from one plane are projected onto another plane from a single point (the center of projection). When this projection occurs between two planes that are parallel to each other, specific geometric properties are maintained or transformed in a predictable way.
Core Transformations
Based on the reference, the most direct effects of this type of central projection are:
- Mapping Lines to Lines: One fundamental feature is that any straight line in the original plane is projected as a straight line in the target parallel plane. This preserves the linearity of geometric figures.
- Scaling Distances: Distances between points are not preserved but are consistently scaled. The reference states that the projection multiplies distances by a constant factor. This means that if you measure a distance in the original plane and its corresponding distance in the projected plane, the ratio between them will be constant for all distances.
Consequences of these Features
Because lines are mapped to lines and distances are scaled by a constant factor, several other important properties are preserved:
- Preservation of Between-ness: If a point B is between points A and C on a line in the original plane, its projection B' will be between the projections A' and C' on the corresponding line in the target plane.
- Preservation of Parallelism: Lines that are parallel in the original plane remain parallel in the projected plane.
- Preservation of Concurrence: Lines that intersect at a single point (are concurrent) in the original plane will have their projections intersect at a single point in the target plane.
These properties are summarized in the reference as consequences of the primary transformations.
Here's a breakdown of the properties mentioned for a central projection between parallel planes:
| Feature | Description |
|---|---|
| Unique Mapping | Each point in one plane corresponds to exactly one point in the other. |
| Maps Lines on Lines | Straight lines remain straight lines. |
| Multiplies Distances | Distances are scaled by a consistent factor. |
| Preserves Between-ness | Relative position of points on a line is kept. |
| Preserves Parallelism | Parallel lines stay parallel. |
| Preserves Concurrence | Intersecting lines remain intersecting. |
The characteristics highlighted in bold, mapping lines on lines and multiplying distances by a constant factor, are explicitly presented as the direct actions of the projection mechanism described in the reference.
