## Assignment 2 2018 – Transformations

Assignment2代写 Throughout this assignment, all sketches that you draw should be carefully drawn to scale using graph paper

**Due: 10Am Friday of week 8: Friday May 4, 2018 10:00Am. Value: 10%. Marked out of 100.**

Throughout this assignment, all sketches that * you *draw should be carefully drawn to scale using graph paper that you can download from the Internet (Exception: your answer to question C1 may be a rough 3D sketch). Use a thinner pen to draw axes and a thicker pen to draw objects so that there is a clear distinction at all times between the object and the axes.Assignment2代写

You will be submitting your solution as a PDF (see below) so ensure that you allow time for scanning your work to prepare your final PDF submission. Also, make sure that your scanned document will be clearly legible – easy to read and clear for marking.

### A. Sketching Transformations in 2D [20 marks]Assignment2代写

A simple triangular flag is sketched below in figure A-1. This shape is deliberately non-symmetric, so it is possible to distinguish between the shape and its mirror image.

Figure A-1: A triangular flag.

A1. [4 marks] Carefully sketch the result of applying the transformation R(90) to the triangular flag. Show the coordinates of the points that define the transformed flag.

A2. [4 marks] Carefully sketch the result of applying the transformation S(2,1) to the original triangular flag. Show the coordinates of points that define the transformed flag.

A3. [4 marks] Carefully sketch the result of applying the transformation R(90)S(2,1) to the original triangular flag. Show the coordinates of the points.Assignment2代写

A4. [4 marks] Carefully sketch the result of applying the transformation S(2,1)R(90) to the original triangular flag. Show the coordinates of the points.

A5. [4 marks] Carefully sketch the result of applying the transformation R(90)T(1,0) to the original triangular flag. Show coordinates of the points.

### B. A Transformation for Stretching in 2D [30 marks]Assignment2代写

A graphics programmer wishes to draw objects that have been stretched in a particular direction. They know how to stretch an object along the X or Y axes, but they need your help to stretch the object in an arbitrary direction. Here are two figures that they provided to help explain what they mean by stretching an object. Figure B-1 shows (a) the original object, which is a circle of radius 1,

(b) the circle stretched by a factor of two along the X axis, (c) the original circle stretched by a factor of 2 along the Y axis, and (d) the original circle stretched by a factor of 2 in a 45 degree direction. Figure B- 2 shows (a) an original unit square and (b) the result of stretching it by a factor of 2 in a 45 degree direction. Note that stretching is about the origin of the axes, not the centre of the object being stretched.

Figure B-1: (a) Original (b),(c),(d) Stretched horizontally, vertically, and diagonally.

Figure B-2: (a) Original (b) Stretched diagonally.

Stretching is defined by a stretch factor 𝑠 and an angleθ, and can be written𝑆𝑡(𝑠, 𝜃). The stretched figures above correspond to:1(b) 𝑆𝑡(2,0), 1(c) 𝑆𝑡(2,90), 1(d) 𝑆𝑡(2,45) and 2(b) 𝑆𝑡(2,45)

#### Answer the following questions:

B1. [3 marks] What well-known transformation corresponds to 𝑆𝑡(𝑠, 𝜃) – i.e. stretching in the X axis direction by a factor of * s*? Your answer should include the parameters of the transformation, and will include at least one mathematical expression that depends on

*.Assignment2代写*

*s*B2. [5 marks] Use the “classic way” to derive a sequence of well known elementary transformations that are equivalent to𝑆𝑡(𝑠, 𝜃). Write the transformations using the abbreviated notation presented in lectures.

B3. [5 marks] Expand each of the elementary transformations in your solution of question B2 to homogeneous matrices. The elements of each matrix will be mathematical expressions involving𝑠 and or 𝜃.

B4. [5 marks] Multiply the homogeneous matrices together to obtain the combined transformation as a single matrix containing mathematical expressions.

B5. [5 marks] Prove that𝑆𝑡(1, 𝜃)is the identity transformation for all values of 𝜃.

B6. [3 marks]𝑆𝑡(−1, 𝜃)is clearly not an identity transformation. Intuitively, what is it? It may help to sketch the result of transforming the unit square B-2(a), or the flag A-1, for some simple values of 𝜃. Mathematically, it may help to consider your answers to questions B1 and B2.Assignment2代写

B7. [4 marks] Apply 𝑆𝑡(2,45) to the four points of the unit square. Show whether the labels on the points in figure B-2(b) are correct or not.

### C. Submarines (50 marks)

A special effects company wants to create a movie scene using Computer Graphics (CG). You are their CG expert. In the movie scene, two submarines cross paths, one above the other.

For this exercise, a submarine is drawn as a cone at each end of a cylindrical body. The submarines are 60 m long overall and 10 m diameter. The nose and tail cones are 5 m long so the cylindrical part of the body is 50 m long. There is a cylindrical conning tower 5 m diameter and 5 m high on top of the submarine. The front edge of the conning tower is 20m from the nose of the submarine.Assignment2代写

Assume that the submarine computer model is aligned with the X axis and the origin is the rear tip of the submarine. World coordinates are expressed in metres.

C1. (4 marks) Draw a rough 3D sketch of the model submarine, including the world axes. This is the * only *sketch in this assignment that you do not need to draw carefully and accurately on graph paper.

C2. (4 marks) Draw an accurate scale drawing of the cross section of the submarine in the plane Z=0. Show all features that intersect the plane.

C3. (35 marks) Suppose that submarine A starts with its origin at the point (1000,0,0). The submarine is to follow a straight path such that its origin passes through the point (0,100,1000).Assignment2代写

#### a.(5 marks) Compute a parametric line equation of the path of the submarine.

b.(10 marks) Assume that the submarine is pointing forwardalong its intended path. Compute the heading and pitch of the submarine A in degrees. You should use a calculator to obtain your solution to two decimal places.

It may be useful to recall the following functions for an angle θ in a right triangle.

sin = opposite / hypotenuse

cos 𝜃 = adjacent / hypotenuse tan 𝜃 = opposite / adjacent.

cos = adjacent / hypotenuse

##### tan = opposite / adjacent.

c.(5marks) Write an abbreviated transformation (using T(…), Rx(…), ) to place and orient submarine A with the heading and pitch that you computed in part b.

d.(5 marks) For each transformation in your answer to part c, write thecorresponding transformation matrix using 3D homogeneous coordinates. You can work to three decimal places for sines and

e.(5 marks) Compute the transformation matrix corresponding to your answer to partc;i.e. compute the product of the matrices that you wrote down in part d. Again, use a calculator and work to three decimal places.Assignment2代写

f.(5marks) How far does submarine A travel from it’s starting point to the other point? Calculate your answer to the nearest metre.

C4. (4 marks) Suppose both submarines travel at the same speed of 72 km per hour (20 m per second). Submarine B travels long the Z axis in a positive direction. What is the starting position of submarine B if it must pass through (0,0,1000) at the same time that submarine A passes through (0,100,1000)?

C5. (3 marks) After crossing above the other submarine, the captain of submarine A instructs a change of heading by 45 degrees while maintaining the same pitch. Show the abbreviated transformation to be applied to the right side of the current transformation matrix to update the heading as required.Assignment2代写

### Submission Assignment2代写

Submit your solutions to this assignment as a PDF file. You may prepare your entire assignment on paper then scan it direct to PDF for submission, or you may scan answers to specific questions and assemble your final submission in a word processing program before converting that to PDF for submission. Both drawings and mathematics may be done by hand on paper and scanned or carefully photographed.Assignment2代写

** Take care **when scanning or photographing your work to ensure that it has good contrast and is clearly legible in the final PDF document. Take care if you are photographing sketches that they are not noticeably distorted – scanning is definitely a better option.

Submit your PDF file through iLearn.

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