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Name: CS354 Email: Midterm Examination 1 Solutions Fall 2003 Each problem section is worth the indicated number of points. There are 5 questions in total that are each worth 20 points. 1. You have discovered (for better or worse) that you are an insect with hexagonal eyes. As a result, you are devising a clipping system for the following hexagonal viewable region: y (4, 3) (4, 1) (5, 0) y = x - 1 y = x - 5 y = -x + 5 (b)[5 ] How would you determine the clipped, visable portion of an arbitrary line in the xy plane? Provide pseudocode. For simplicity, you may assume that the line does not coincide with any of the lines defined in Part (a). Test the arbitrary line for intersection against all 6 clipping lines in Part (a) by solving 6 systems of linear equations. For each clipping line, we know (x , y ) and (x , y ), which are the endpoints of the visible region of that clipping line. Test if the intersection point for a clipping line is between its two endpoints, and if so, record the point. When all 6 intersection tests are complete, remove any duplicate points that were recorded and return the remaining points. If two points are returned, a segment of the arbitrary line passes through the viewable region. If one point is returned, the arbitrary line is tangent to the viewable region. If zero points are returned, the arbitrary line does not pass through the viewable region. (c)[10 ] Does any part of the line appear in the viewable region? If so, find the endpoints of the visible line segment. Be sure to show your work. Using the algorithm above, the points that are recorded are (4,2) from the clipping line and ( , ) from the clipping line . These two points are unique, so neither is removed. The endpoints of the visible line segment are (4,2) and ( , ). 1 ( ' % &$ ( ' " #! 3 4210) (a)[5 ] Give the or form of all line equations that will be used for clipping. y = -x + 9 x = 4 x = 6 (5, 4) (6, 3) (6, 1) x 2. A 2-D line segment can be expressed in parametric form as: We can reuse the original parameterization of the line segment that we are given, except the old parameter t on the range [0,1], needs to be linearly transformed to the new parameter s on the range [-1,1]. Since the transformation is linear, it must be of the form: Use the min and max for the range of s and the min and max for the range of t to determine the constants a and b: Min coordinates: Max coordinates: Substituting in: Min coordinates: Max coordinates: Solving, we find and . Thus: We can plug this formula for t into the original parametric equations to obtain a parametrization of the original line segment in terms of s. % % 3 ( 2 ( & 2 1100 ) ' ( ( & Simplifying: 2 (a)[10 ] Give a parametric representation for a line segment in terms of a new parameter such that when the endpoint is specified, when , the endpoint is specified, and when midpoint of the line segment is specified. ' where the endpoints of the line segment are and varies from 0 to 1, all points on the line segment starting from respectively. In this form, as the parameter and ending at are specified. , the ' 2 $#" #"! !#" !#" & & 2 & ! (b)[10 ] Give a parametric representation for a circle of radius centered at in terms of a parameter such that when the leftmost point on the circle is specified, when the topmost point is specified, when the rightmost point is specified, and when the bottommost point is specified. Hint: The parametric representation of a unit circle centered at the origin, starting from (1,0) and going in the counterclockwise direction is: Multiply the right side of the paramtric equations for a unit circle by r to scale the unit circle to a circle with radius r. Add an offset of x and y to x and y, respectively, to translate the circle center from (0,0) to (x , y ). Finally, replace the old parameter t with the new parameter ( ). The offset of indicates the new parameter starts at the leftmost point in the circle and the factor of - indicates that the new parameter moves clockwise and steps in increments of for every unit that the parameter is increased. The new parameter produces the desired clockwise traversal of the circle. x = x + r cos(- t + ) y = y + r sin(- t + ) 3 " where . " & 3. Consider the following 2-D affine transformation (the dashed lines represent an imaginary box for your reference - this box is NOT part of the image): 3 6 8 -5 (a)[15 ] Specify a sequence of basic affine transformations that can be applied to the left image to generate the right image. You may use the following three types of affine transformations: Scaling [ S_x [ 0 [ 0 0 S_y 0 0 ] 0 ] 1 ] Rotation (counterclockwise about origin) [ cos(ang) -sin(ang) 0 ] [ sin(ang) cos(ang) 0 ] [ 0 0 1 ] Translation [ 1 [ 0 [ 0 0 1 0 T_x ] T_y ] 1 ] Be sure to indicate the the order in which the should transformations be applied. Correction: In the original midterm, the rotation matrix above was labeled as a "clockwise rotation." The label should have read "counterclockwise rotation." No points were deducted if you assumed either of the two orientations. We will perform the transformation as follows, using the transformation of points technique (rather than the turtle graphics technique): 1. Translate the center of the face to the origin, using a translation of (-7,4). 2. Perform a counterclockwise rotation of -90 degrees, which is a clockwise rotation of 90 degress. Since the face is centered at the origin and since rotations always take place about the world origin (when we are applying transformations to points in a scene), the face will be rotated about its center. Note that in turtle graphics, we have a different situation where the object coordinate axes are rotated about the object coordinate origin. 3. Translate the center of the current face to the center of the face in the right figure above. [ [ [ 1 0 0 Step 1 0 -7 1 4 0 1 ] ] ] [ 0 [ -1 [ 0 Step 1 0 0 2 0 0 1 ] ] ] [ [ [ 1 0 0 Step 0 1 0 3 4 2 1 ] ] ] (b)[5 ] Use your answer from Part (a) to find a single transformation matrix that can be applied to the left image to generate the right image in one step. Show your work. Since we are using the transformation of points technique, new matrices are multiplied to the left side of existing matrices, which will reverse the left-to-right order in which the matrices appear in Part (a). [ [ [ 1 0 0 0 1 0 4 2 1 ] ] ] x [ 0 [ -1 [ 0 1 0 0 0 0 1 ] ] ] x [ [ [ 1 0 0 0 -7 1 4 0 1 ] ] ] = [ 0 [ -1 [ 0 1 0 0 8 9 1 ] ] ] 4 -3 x 1 x 5 3 y y 4. You are given the following fractal, which is a unit square on the 0th iteration: Initial Shape On each iteration: 1 (a)[10 ] What is the fractal dimension? Show your work. Hint: If on each iteration each piece of length x is replaced by b nonoverlapping pieces of length x / a, then the fractal dimension is . Notice that on each iteration, each edge in the fractal is replaced by 11 nonoverlapping edges that are the length of the original edge. Edges that appear two times as long as the other edges are really composed of two edges of equal length. (b)[10 ] Assume the turtle begins in the lower left corner of the square and is pointing up. Write an L-system that represents the fractal. For each L-system symbol that you use, be sure to clearly define what it represents (ex: draw line, turn, etc.). 'F' '+' '-' 's' '[' ']' - draw a unit line segment (i.e. a line segment of length 1) turn 90 degrees right turn 90 degrees left scale x and y by a factor of 1/5 push pop Axiom: F + F + F + F Rule: F -> s[F - F + F + F F - F - F F + F + F - F] 5 1 3% 2 5. Imagine being given the following 5 control points, which define two Bezier curves. The first curve consists of the control points , , and , while the second curve consists of the control points , , and . P1 (s, 5) P4 (u, 3) P0 (-9, 0) P3 (2, t) (a)[10 ] Two adjoined Bezier curves and have geometric continuity holds at their common point P if the first derivative of with respect to t, evaluated at P, is a constant multiple of the first derivative of with respect to t, evaluated at P. If there is continuity at Point , what do we know about the values of s and t? You should express t as a function of s. Start from the definition of continuity and show your work. Solving the x coordinate equation for c: 6 We evaluate the x and y coordinates separately. Plugging in the x and y coordinates for x coordinate: y coordinate: " " " $ $ %! $ " " "" & & Let c be some constant. By , so: $ $ "" " 1 " " ! " # " & # #" " !& "" Evaluating at , where (t = 0): " Similarly, for the right Bezier curve % " " " # "" " ! Evaluating at % " " " ! For the left Bezier curve : , where (t = 1): : geometric continuity, we know that at their common point , , and : ! P2 (1,1) Hint: Remember that a Bezier curve is defined from three control points . , , as: ! ! " " " " 1 Plugging this result into the y coordinate equation and simplifying: Finally, solve this equation for t: The variable u does not appear in any equation related to continuity at . We therefore know nothing about the value of u. Also, because u does not appear in any equation related to continuity at , its value does not affect continuity. The curves would still have continuity if the value of u were doubled. 7 (b)[10 ] If there is continuity at Point doubled, would the curves still have " 1" & ( ! , what do we know about the value of u? If the value of u were continuity?

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