CS 3744, Assignment 6 - Due April 8, 2010

Please show all your work!

Consider in this homework two directed line segments L1 and L2 as follows:

L1: (10, 20) to (-40, 10)
L2: (-10, 5) to (10, 10)

  1. Express infinite lines L1 and L2 in slope-intercept form

  2. Express infinite lines L1 and L2 in point-slope form

  3. Express infinite lines L1 and L2 with the r, θ parameterization.

  4. Express infinite lines L1 and L2 in vector dot product form, and give the normalized (unit) normal vectors for each.

  5. Express finite lines L1 and L2 in parametric form and determine the coordinates that correspond to parameter alpha = -0.5, +0.4, and +1.2 for the parametric representation you choose.

  6. Express the endpoints of L1 in homogeneous coordinates.

In the following, let V1 be the vector from the first point of L1 above to the second point and let V2 be the vector from the first point of L2 to the second point.

  1. Express the vector V2 in homogeneous coordinates.

  2. Determine the angle between -V1 and V2, the length of the projection of V2 onto -V1, and the normalized unit vectors corresponding to V1 and V2.

  3. Assuming the vectors are in the z = 0 plane, determine -V1 × V2, determine its Euclidean length, and a determine a corresponding unit vector.

  4. Determine the distance of the point (1, 6) from the line y = 2x - 2 and determine the location on the line of the projection of this point.

  5. Let u = (a, b) and v = (c, d) be members of a 2D vector space and and consider an operation (u, v) on this vector space defined by (u, v) = ac. Show whether or not (u, v) is an inner product.

  6. Suppose we have a convex polygon P with vertices (2, 4), (6, -1), (-2, -4), and (-4, -1). Set up the matrix method shown in class for testing whether arbitrary points R = (x, y) are strictly inside P or not. Use this formulation to determine whether the point (3, -2) is inside or not, and determine the Euclidean distance of (3, -2) from each of the edges bounding the polygon P.

  7. Suppose we are given the two finite line segments from (-4, 3) to (2, -3) and from (-2, -4) to (6, -1). Use the parametric formulation given in class to determine whether these two finite line segments intersect. Then use the parametric representation to determine the intersection point of the infinite line segments.

  8. Below are two copies of a complex polygon. Shade one with the winding rule and the other with the parity rule (label each).

            

  9. Suppose a plane is given by the equation x + 5y + 2z - 6 = 0.

    (a) Determine a point-normal form of the equation of this plane.

    (b) Determine a parametric form for the equation of this plane.

  10. Suppose we are given a triangle with different intensities at the vertices as follows:

    (6, 12) Intensity = 40
    (12, 0) Intensity = 80
    (0, 8) Intensity = 50
    Using bilinear interpolation on a horizontal line through the point P = (7, 6), determine the interpolated intensity at point P.

  11. Given the line equation (10, -2, 4)·(x, y, 1) = 0,

    (a) What is the difference between a line and a vector?

    (b) Determine a vector perpendicular to this line.

    (c) Determine a vector parallel to this line.

  12. Given the three points in 3D (1, 0, -4), (1, 1, 0), and (6, -8, 2), give a parametric equation of the plane containing these three points.

  13. Suppose we are given the 3 × 3 matrix {{-1, 2, 3},{1, 1, 1},{4, -2, -1}}. calculate the determinant and give the rank.

  14. Consider the two finite line segments from (-8, 11) to (3, 2) and from (-1, 3) to (8, 10). Determine the values of t and u of the intersection point and determine the (x,y) coordinates where the two lines intersect.
This assignment is due in class on the date above.