a) Transform the quadrilateral ABCD with vertices A(1,0),B(4,−1),C(5,3) and D(−1,5) under a translation by the point (4, 5) followed by a counter-clockwise rotation by an angle of 45◦
.b) If you perform an x-direction shear transformation, and then a y-direction shear transformation, will the result be the same as the one which is obtained when it is simultaneous shear in both the directions? Justify your answer.
c) Let W be a window with corners (0,0),(8,0),(8,4) and (0,4). Clip a triangle with vertices (1,1),(10,2) and (5,9) against the window W by tracing Liang Barskey line clipping algorithm
Let's break down each part of your question step by step: ### Part (a): Transform the quadrilateral under translation and rotation We are given the vertices of the quadrilateral: \( A(1,0), B(4,-1), C(5,3), D(-1,5) \) The transformations to be applied are: 1. Translation by the point (4, 5) The translation formula for a point \( P(x, y) \) by a vector \( (T_x, T_y) \) is: \[ P' = (x + T_x, y + T_y) \] Substituting the translation vector \( (4, 5) \), we apply this to each point: - \( A(1, 0) \to A'(1+4, 0+5) = A'(5, 5) \) - \( B(4, -1) \to B'(4+4, -1+5) = B'(8, 4) \) - \( C(5, 3) \to C'(5+4, 3+5) _______ _________ ________ _________ _______ __________ _______ _____ ______.
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a) Devise an efficient algorithm that takes advantage of symmetry properties to display a sine function.
b) Prove that the reflection along the line y = −x is equivalent to reflection along the y-axis followed by a counter-clockwise rotation by 90◦ .
c) Shear a square whose opposite vertices are at (1,1) and (2,2) by i) 2 units along the x-axis and reference line y = 0
ii) 4 units along the y-axis and reference line x = 0.
a) Consider a polygon with vertices at (5, 20), (12, 5), (15, 15), (25, 5), (30, 25), and (15, 30). Prepare a sorted edge list, and then make the active edge list for the
scanlines y = 5,10,15,20,25,30.
b) Develop and implement the flood fill algorithm.
c) Prove or disprove: "Multiplication of transformation matrices for two successive rotations is commutative."
a) Transform the quadrilateral ABCD with vertices A(1,0),B(4,−1),C(5,3) and D(−1,5) under a translation by the point (4, 5) followed by a counter-clockwise rotation by an angle of 45◦
.b) If you perform an x-direction shear transformation, and then a y-direction shear transformation, will the result be the same as the one which is obtained when it is simultaneous shear in both the directions? Justify your answer.
c) Let W be a window with corners (0,0),(8,0),(8,4) and (0,4). Clip a triangle with vertices (1,1),(10,2) and (5,9) against the window W by tracing Liang Barskey line clipping algorithm
a) If the origin is taken as the centre of projection, then what will be the perspective projection when the projection plane passes through the point P(4,5,3) and has normal vector (1,2,−1).
b) Write a program that produces different views of a cuboid, that is, how the cuboid looks from the top, from the front or from the right.
c) Write a code to continuously rotate a pentagon about a corner point in the anti-clockwise direction.
a) Write a boundary fill procedure to fill an 8-connected region.
b) Let W be the window having two diagonally opposite corners at (10, 2) and (30, 15). Trace the Cohen-Sutherland line clipping algorithm for the line segment joining the points (0,0) and (15, 30).
a) Explain what do you understand by the terms persistence, refresh rate, resolution, aspect ratio, horizontal and vertical retrace.
b) Compute the pixel positions along the line path of the lien joining the points A and B(10,8).
c) Using the midpoint method and symmetry in account, develop an efficient method
for scan converting the curve n the interval [0, 10].
a) What is the difference between a parallel projection and a perspective projection? Explain with examples.
b) What will be the perspective projection of a unit cube on the plane x = y if it is viewed from the point (1,2,0)? Justify your answer.
c) Transform the scene in the world coordinate system to the viewing coordinate system with viewpoint at (1,1,2). The view plane normal vector is(−4,2,5) and the view up vector is (1,4,0).