The projection of a line segment on a plane is the line segment formed by connecting the projections of its endpoints onto that plane.
In geometry, understanding projections helps visualize how 3D objects appear when "flattened" onto a 2D surface like a plane. For a simple element like a line segment, the concept is quite straightforward, following the fundamental principle of projecting points.
Understanding Projection
Projection, in this context, involves casting a "shadow" of an object onto a surface using parallel rays. For orthogonal projection (which is typically assumed unless otherwise specified), these rays are perpendicular to the surface.
Projection of a Point
The projection of a single point onto a plane is the foot of the perpendicular line drawn from the point to the plane. Imagine dropping a point straight down onto the plane; where it hits is its projection.
Projection of a Line Segment
Building upon the projection of points, the projection of a line segment follows a similar logic, analogous to how a line segment is projected onto another line:
- Reference Principle: Just as "The projection of a line segment joining two points on a line is the line segment, formed by taking the projection of the endpoints of the parent line segment on the existing line," the same principle applies to projecting onto a plane.
- Applying to a Plane: If you have a line segment connecting point A and point B in space, its projection onto a plane is found by:
- Finding the projection of point A onto the plane (let's call it A').
- Finding the projection of point B onto the plane (let's call it B').
- The projection of the line segment AB on the plane is the line segment A'B'.
This projected segment A'B' lies entirely within the plane.
Characteristics of the Projected Segment
The nature and length of the projected line segment A'B' depend on the orientation of the original line segment AB relative to the plane.
- Parallel Segment: If the line segment AB is parallel to the plane, its projection A'B' will be a line segment equal in length to AB.
- Perpendicular Segment: If the line segment AB is perpendicular to the plane, its projection A'B' will be a single point (the projection of both A and B coincide).
- Angled Segment: If the line segment AB is at an angle (θ) to the plane, its projection A'B' will be a line segment with a length equal to the length of AB multiplied by the cosine of the angle (Length(A'B') = Length(AB) * cos(θ)). The angle θ is the angle between the segment and the plane.
Visualizing the Projection
Imagine a flashlight held far away (casting parallel light) shining directly down (perpendicularly) onto a table (the plane).
- Hold a pencil flat on the table: its shadow is the same length. (Parallel case)
- Hold a pencil pointing straight down at the table: its shadow is just a point. (Perpendicular case)
- Hold a pencil at an angle above the table: its shadow is shorter than the pencil. (Angled case)
This shadow represents the orthogonal projection of the pencil (line segment) onto the table (plane).
Summary
The projection of a line segment onto a plane is determined solely by the projections of its two endpoints onto that plane. This results in another line segment on the plane (or a single point in the specific case where the original segment is perpendicular to the plane).
