Computer graphics Assignment 1

Geometry Review
Due date: Wednesday, September 22, before class

In all cases, the direction and perpendicular vectors of the lines and normals to the planes are assumed to have unit length. The problem marked with * is optional; extra credit will be given for it. Providing an answer is insufficient; you have to explain how you have arrived at the answer.

1.

Determine which of the following pairs of lines intersect, and find the intersection points; explain your calculations.

(a)

the line with perpendicular [-3,1] passing through [2,1], and the line x + y = 12;

(b)

the line with direction [1,2] intersecting the y axis at 3 and the line with equation 2x - y + 5 = 0;

(c)

the line with direction [1,1] passing through [0,1] and the line with normal [1,-1.0001] passing through the origin.

2.

Find the expression for the distance from a point p₀ to a line with normal n passing through the point p.

3.

Derive the parametric and implicit equations of the line passing through two given points p₁ and p₂.

4.

Derive an equation that has to be satisfied by three points p₁, p₂ and p₃ which are collinear, that is, lie on the same line.

5.

Determine when a line with direction v passing through point p intersects the ellipse x²/a² + y²/b² = 1. In the cases when there is an intersection, find the expressions for intersection points.

6.

Derive the implicit equation of the plane passing through points p₁, p₂ and p₃, when such plane is unique. State the condition on p₁, p₂ and p₃ for the plane to be unique.

7.

Given two planes with normals n₁ and n₂ passing through points p₁ and p₂ respectively, find the parametric equation for their line of intersection, if any.

8.

Find the angle between a line in space with direction v and the plane with normal n.

9.

Find the parametric equation of the line satisfying the following three conditions:

(a)

it is parallel to a plane p with normal n passing through a point p₁;

(b)

it is perpendicular to the line l with direction v passing through the point p₂;

(c)

it passes through the intersection point of the line l and the plane p.

10.

^* Find the distance between two lines p₁ + v₁ t and p₂ + v₂ t in space.