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- Computer Graphics Assignment 4: Transformations and Barycentric Coordinates

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Exercise 1 [5 points]

Let us consider a 2D coordinate system with points p1 = (1, 1)T

, and p = (1.5, 2.5)T

. Additionally, let’s define

a vector u = p − p1. Perform the following tasks:

Task 1 [1 points] Construct two matrices, R90, T ∈ R3×3

, such that the first one performs counter-clockwise

rotation around the center of the coordinates system by angle 90◦

, and the second one performs translation

by vector t = (1, −2)T

, in homogeneous coordinates.

Task 2 [1 points] Represent the points p1 and p as well as the vector u using homogeneous coordinates and

transform them first using the R90, and then T matrices. Convert the obtained points and vector back into

Cartesian coordinates and denote them by p

0

1

, p0

, and u

0

. Draw all the points and vectors before and after the

transformation, and verify that u

0 = p

0 − p

0

1

. What influence did the matrix T have on the u

0

?

Task 3 [1 points] Construct a scaling matrix, S ∈ R3×3

, which scales everything in x and y direction by a

factor of 2. Compute new points p

00

1

, p00 and vector u

00 by transforming p

0

1

, p0

, and u

0 using matrix S. Again

perform the operation in homogeneous coordinates, and then, convert the results to Cartesian coordinates

and visualize them.

Task 4 [2 points] Construct matrices R

−1

90 , T −1

, S−1 ∈ R3×3

, which perform inverse transformations to

R90, T, S,, i.e., clockwise rotation by 90◦

, translation by vector t

0 = (−1, 2)T

, and scaling by factor 0.5. Compute matrix M = R

−1

90 T

−1S

−1

, and verify that transforming points p

00

1

, p00, and u

00 with this matrix results in

the initial values of p1, p, and u .

Exercise 2 [4 points]

Consider a triangle made of three points p1 = (6, 0, 4), p2 = (2, 0, 0), and p3 = (2, 4, 4). Verify whether point

p = (4, 1, 3) belongs to the interior of the triangle.

Exercise 3 [3 points]

Using barycentric coordinates, prove that the centroid of a triangle divides its medians in ratio 2:1.

Exercise 4 [3 points]

Consider a transformation that projects all the points pi ∈ R2 onto

the line x = 1. For each point, the projection is performed along

the line that passes through the center of the coordinate system

and the point (see the image on the right). Each point p

0

i

is a

result of applying this transformation to point pi

. Construct a

matrix M ∈ R3×3 which realizes this transformation using homogeneous coordinates. More specifically, after converting a point

pi

into the homogeneous coordinates, multiplying it with the matrix M, and transforming it back to the Cartesian coordinates, you

should obtain point p

0

i

. Note that this is not an affine transformation. Comment on how the matrix M transforms the points lying

on the y-axis. Interpret the results both in the homogeneous and

Cartesian coordinates.

Bonus exercise 5 [2 points]

Consider the same task as in Exercise 4, but the line onto which the points are projected can be now arbitrary,

and it is defined by a line equation y = ax + b, where a, b ∈ R are constants. Derive the matrix M for this

more general case.

Submission

Submit a single PDF including all the calculations you did to solve the assignments to iCorsi. You should not

use any external tools, other than a simple calculator, which can provide intermediate or final results.

Solutions must be returned on October 20 2022 via iCorsi3

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