Show that the image of an ellipse in a plane, not necessarily one parallel to the image plane, is also an ellipse. Show the the line in space is a line in the image. Assume perspective projection.

To show that the image of an ellipse in a plane is also an ellipse when projected onto a different plane using perspective projection, and to show that a line in space is a line in the image, we can use mathematical reasoning and geometric principles.

Let's start by considering the perspective projection. In perspective projection, parallel lines in space converge to a vanishing point on the image plane. This means that if we have an ellipse in space (lying in a plane), its projection onto another plane will also have the same vanishing point.

Now, let's mathematically prove that the image of an ellipse is also an ellipse.

1. Suppose we have an ellipse in space represented by the equation:
(x/a)^2 + (y/b)^2 + (z/c)^2 = 1,
where a, b, and c are the semi-axes of the ellipse.

2. Let's assume that the center of projection and the image plane are aligned such that the center of the ellipse lies on the xy-plane (z = 0), and the projection plane is parallel to the xy-plane.

3. Now, let's consider a point (x, y, z) on the ellipse. When the point is projected onto the image plane, the x and y coordinates remain the same, while the z-coordinate becomes zero.

4. The projected point is thus (x, y, 0).

5. Plugging this projected point into the equation of the ellipse, we get:
(x/a)^2 + (y/b)^2 = 1.

6. The resulting equation is the equation of an ellipse in the image plane. Hence, the image of an ellipse is also an ellipse.

Now, let's move on to proving that a line in space is also a line in the image.

1. Assume we have a line in space represented by the parametric equations:
x = x₀ + at,
y = y₀ + bt,
z = z₀ + ct,
where (x₀, y₀, z₀) is a point on the line and (a, b, c) is the direction vector of the line.

2. When this line is projected onto the image plane, the x and y coordinates remain the same, while the z-coordinate becomes zero.

3. The projected line becomes x = x₀ + at, y = y₀ + bt, and z = 0.

4. Eliminating z from the equations, we have:
x = x₀ + at,
y = y₀ + bt.

5. These equations represent a line in the image plane, which means a line in space is also a line in the image.

In conclusion, when an ellipse in space is projected onto another plane using perspective projection, and when a line in space is projected onto an image plane, both remain ellipses and lines, respectively, in the image plane.