Exercises

Assignment Details

• Assignment Link: https://weblab.tudelft.nl/cse2215/2024-2025/assignment/131897/
• Pass Criteria: Pass All Sub-assignments (How many weighted sub-assignments should one pass to pass this folder?)
• Available on: Mon, Nov 11, 2024 23:59:00
• Weight: 1.0 of total 6.0
• Minimum to Pass: 6.0
• Due: Fri, Nov 22, 2024 23:59:00
• Deadline: Fri, Nov 22, 2024 23:59:00

Description

Linear Algebra Recap (Part of Assignment 1)

This recap test comprises multiple-choice (MC) and open questions with formatted answers. This assignment helps you recap the linear algebra content necessary to master the learning goals in the “Computer Graphics” course. The exercises correspond to the content covered in the linear algebra recap lecture.

Instructions:

• The folder “Ungraded Linear Algebra Recap Assignments” contains assignments that do not count towards the final grade of this recap. You are still invited to fill these in to fully test your mastery of the subject.
• Both MC and open questions are worth 2 points.
• For MC questions, 2 points are added to your total for a correct answer; 1 point is added if you choose “Skip Question” to compensate for partial answers; no points are added if you answer incorrectly.
• For the open questions, no point is added if you skip the question (leave it blank). For a perfect answer, you similarly receive (as for MC questions) 2 points. No points are added for incorrect answers or partially correct answers.
• In the midterm, a simple, non-graphical calculator is permitted, therefore, we suggest only using such for this assignment.

Sub-assignments

Progress State Explanation:

• Started State: For questions (e.g. Programming, Multiple Choice and Essay Questions), submissions are marked started once students save their submission for the first time.
• Completed State: For questions (e.g. Programming, Multiple Choice and Essay Questions), a submission is marked completed when it is submitted. A submission gets submitted by the student (by hitting submit), or automatically after a deadline passes.
• For folders, a submission is marked completed only when the number of completed child submissions meets the pass criteria (Pass All or Pass n-of-m*) of the folder.
• *Pass All and Pass n-of-m will only consider non-zero weight assignments, or all assignments when all have zero weight.
• Started/Completed state at the folder level is updated periodically, i.e. updates may not be reflected immediately.

Result State Explanation:

• Grades: Grades will only be shown when they are published by course staff. Grade calculations can be viewed by clicking on the grade.
• Passed State: For questions (e.g. Programming, Multiple Choice and Essay Questions) and folders, a submission is marked passed, when the (computed) grade is higher than the minimum to pass.
• In addition, for folders, the number of passed child submissions should also meet the pass criteria (Pass All or Pass n-of-m*) of the folder.
• *Pass All and Pass n-of-m will only consider non-zero weight assignments, or all assignments when all have zero weight.
• This figure is computed periodically, i.e. updates may not be reflected immediately.

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In $\mathbb{R}^2$, the addition of two vectors can be visualized by attaching the endpoint of one vector to the origin and the endpoint of the other to the tip of the first. Visualize the addition $\mathbf{v}+\mathbf{w}$ with $\mathbf{v}=\binom{1}{2}$ and . Verify the result of this addition by calculating the answer.

Options:

• $\binom{-1}{7}$
• $\binom{4}{7}$

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Description

Compute $\mathbf{v}-\mathbf{w}$ with $\mathbf{v}=(-1,0,3)^t$ and $\mathbf{w}=(-2,$.

Options

Select one option

• $(4,-4,6)^t$
• $(-3,-4,3)^t$

Discussion with Course Staff

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Addition of Vectors (2D)

Given two vectors $\mathbf{v}$ and $\mathbf{w}$, calculate the vector $\mathbf{v}+\mathbf{w}$.

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Subtraction of Vectors (2D)

Given two vectors $\mathbf{v}$ and $\mathbf{w}$, calculate the vector $\mathbf{v}-\mathbf{w}$.

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Description

Given two vectors $\mathbf{v}$ and $\mathbf{w}$, calculate the vector $\mathbf{v}+\mathbf{w}$.

Answer

Discussion

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Description

Given two vectosr $\mathbf{v}$ and $\mathbf{w}$, calculate the vector $\mathbf{v}-\mathbf{w}$.

Solution

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Length of a Vector (MC)

What is the length of the vector $\mathbf{v}=\binom{1}{2}$?

Select one option

• $\sqrt{5}$
• $\sqrt{3}$

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Description

Compute the length of the vector $\mathbf{v}$.

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Description

Compute the length of the vector $\mathbf{v}$.

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Question

Determine the dot product between $\mathbf{v}=(-2,4)^t$ and $\mathbf{w}=(1,0)^t$.

Select one option

• -2
• 7
• 3
• $\sqrt{15}$

Assignment Details

• Assignment Link: https://weblab.tudelft.nl/cse2215/2024-2025/assignment/131918/
• Available on: Mon, Nov 11, 2024 23:59:00
• Weight: 1.0 of total 16.0 weight of assignment
• Minimum to Pass: Not set
• Due: Fri, Nov 22, 2024 23:59:00 advised date to finish this assignment
• Deadline: Fri, Nov 22, 2024 23:59:00 final date and time an assignment can be completed (without penalty)

Description

Given the vectors $\mathbf{v}$ and $\mathbf{w}$, compute the dot product $\mathbf{v}\cdot\mathbf{w}$.

Description

Given the vectors $\mathbf{v}$ and $\mathbf{w}$, compute the dot product $\mathbf{v}\cdot \mathbf{w}$.

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Angles (MC)

Calculate the angle between the vectors $\mathbf{v}=\binom{-1}{1}$ and $\mathbf{w}=\binom{0}{-1}$ using the dot product.

• $\frac{3\pi}{4}$
• $\frac{\pi}{2}$

Discussion with Course Staff

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Distance between two Points (MC)

What is the distance between points $\mathbf{v}=\binom{-1}{0}$ and point $\mathbf{w}=\binom{2}{-1}$? Visualize the distance and verify it by computing it.

Select one option

• $4$
• $\sqrt{4}$
• $10$
• $\sqrt{10}$

Assignment Details

• Assignment Link: https://weblab.tudelft.nl/cse2215/2024-2025/assignment/131912/
• Available on: Mon, Nov 11, 2024 23:59:00
• Weight: 1.0 of total 21.0
• Minimum to Pass: Not set
• Due: Fri, Nov 22, 2024 23:59:00
• Deadline: Fri, Nov 22, 2024 23:59:00
• Submission of: Student
• Assignment Deadline (by parent collection): Fri, Nov 22, 2024 23:59:00

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Linear Independence (MC)

Select one option:

1. Are the vectors $\mathbf{u}=(1,2,1)^t$, $\mathbf{v}=(1,3,1)^t$, linearly independent?

Possible Answers:

• $\mathbf{u},\mathbf{w},\mathbf{v}$ are linearly independent, but $\mathbf{a},\mathbf{b},\mathbf{c}$ are not.
• $\mathbf{u},\mathbf{w},\mathbf{v}$ are not linearly independent, but are.

Discussion with Course Staff

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Given a vector $\mathbf{v}$, find vector $\mathbf{w}$ such that $(\mathbf{v,w})$ is a basis of $\mathbb{R}^2$.

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Description

Building a Basis (3D)

Given a vector $\mathbf{v}$, find vectors $\mathbf{u,w}$ such that $(\mathbf{u,v,w})$ is a basis of $\mathbb{R}^3$.

Given the vector $\mathbf{v}=\binom{1}{2}$, compute a vector $\mathbf{w}$ orthogonal to it. Then, $\mathbf{w}$ is of the following form:

Select one option

• $\mathbf{w}=r\binom{-2}{4}$ for some scalar $r$.
• $\mathbf{w}=r($ for some scalar .

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Description

Given a vector $\mathbf{v}$, provide a vector $\mathbf{w}$ that is orthogonal to $\mathbf{v}$.

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Orthogonal Vector (3D)

Given a vector $\mathbf{v}$, provide a vector $\mathbf{w}$ that is orthogonal to $\mathbf{v}$.

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Inverse of a Matrix (MC)

Given the matrix $\begin{pmatrix}1 & 2\\ 3 & 4\end{pmatrix}.$ Which of the following is an inverse of the matrix?

Select one option

• $\begin{pmatrix}-2 & 1\\ \frac{3}{2} & -\frac{1}{2}\end{pmatrix}$

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Description

Inverse of a Matrix (3D)

Given a matrix with column vectors $\mathbf{u}=(u_1,u_2,u_3)^t$, and , i.e., the matrix

calculate its inverse.

Solution

Answer:

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Description

Line 2D (MC)

Which of the following vector representations does correspond to the line given by the equation $y=-\frac{1}{2}x+1$?

Options

Select one option

• $\binom{0}{0}+s\binom{1}{-\frac{1}{2}}$

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Line 3D (MC)

Which of the following formulations corresponds to the line in $\mathbb{R}^3$ that connects points (1,1,0)^t and (0,0,2)^t.

Select one option

• $(\frac{1}{2},\frac{1}{2},1)^t + r(1,1,-2)^t$

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Points on a Line (MC)

Given the line $(1,2,3)^t + s(0,2,4)^t$ in ${\mathbb{R}}^3$, which of the following points are on that line: ?

Select one option

• $u,w,x$ are on the line.
• $u,v,y$ are on the line.
• $w,x,y$ are on the line.
• $$ are on the line.

Assignment Details

• Assignment Link: https://weblab.tudelft.nl/cse2215/2024-2025/assignment/131933/
• Available on: Mon, Nov 11, 2024 23:59:00
• Weight: 1.0 of total 21.0
• Minimum to Pass: Not set
• Due: Fri, Nov 22, 2024 23:59:00
• Deadline: Fri, Nov 22, 2024 23:59:00

Submission Details

• Submission of: Student
• Assignment Deadline: Fri, Nov 22, 2024 23:59:00 (by parent collection)

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Normal form (MC)

Which of the following normal forms describes the plane \((0,1,2)^t+s(1,2,3)^t+r(4,3,2)^t\)?

• \((-5, 10, -5)\cdot ( \mathbf{v} - (-4,13,0)^t) = 0 \)
• \((1, 2, 3)^t \cdot (\mathbf{v} - (1,3,5)^t) = 0\)
• \((4, 3, 2)^t\cdot (\mathbf{v} - (-4,13,0)^t) = 0\)
• \((-5, 10, -5)^t\cdot (\mathbf{v} - (1,3,5)^t) = 0\)
• Skip Question

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Vector Form (MC)

Which of the following vector forms does correspond to the plane given by normal form $(1,2,3)^t \cdot (v-(0,0,1)^t) =0$?

Select one option

• $(3,0,0,)^t + s(3,0,0)^t + r(\frac{1}{2},0,-1)^t$

Discussion with Course Staff

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Points on sides of a Plane (MC)

Given the plane $(0,0,0)^t + s(1,2,3)^t + r(2,3,4)^t$, determine which of the points lie on the same side of the plane.

Select one option

• All lie on the same side.
• $u$ on one side, $v,w$ on the other.
• $v$ on one side, $u,w$ on the other.
• $w$ on one side, on the other.

Discussion with Course Staff

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Description

Given two points $\mathbf{u},\mathbf{v}\in\mathbb{R}^2$, compute the mid-point $\mathbf{b}$ between the two points and the direction $\mathbf{s}$ of the separating line between the points that is orthogonal to the line connecting $\mathbf{u}$ and .

Answer

(The answer section is empty in the provided HTML.)

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Description

Starting from the point $a=(3,0,3)^t$, a line is shot in direction $(-1,1,-0.5)^t$. It hits the -plane at point , where it is reflected such that the angle of incidence is equal to the angle of reflection. It then hits the -plane at point , see the (not to scale) depiction below.

Which of the following statements on the coordinates of $b$ and $c$ is correct?

Options

• $\mathbf{b}=(0,3,1.5)^t$, $\mathbf{c}=(3,6,0)^t$

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The following code draws an image by setting the pixel colors. Which of the following images corresponds to the output of this code? (the mod(x,y) function returns x mod y)

bool PixelColor (int coordx, int coordy) {

  int x = mod(coordx, 10);
  int y = mod(coordy, 8);

  if (y == 0 || y == 4)
    return false;
  if (y < 4 && x == 4)
    return false;
  if (y > 4 && x == 9)
    return false;
  return true;
}
void CreateImage () {
  for (int i=0; i<20; ++i)
    for (int j=0; j<20; ++j)
    {
      if (PixelColor(i,j))
        Img.pixel(i,j) = (1.0, 0.0, 0.0);
      else
        Img.pixel(i,j) = (, , );
    }
}

• 

• 

• 

• Skip Question

• 

• 

• 

• 

Transformations

You are asked to simulate a 2D bicycle, as illustrated below. The centre of the bicycle $c$ is initially at position $(0,0)$ and the centre of the crank is at position $a=(1, -2)$. The centre of the pedal $b$ rotates around point . For every complete clockwise turn of the pedal, the bike moves unit forward (positive x-axis).

Given the initial position $b=(2.5,-3)$, compute the position $b$ of the pedal after a rotation of $\theta=-\frac{\pi}{2}$.

Write your answer in the form $(b_x, b_y)$. For example, a possible answer could be:

$3.3, -5.55$

Check your answer with the following information: $b_x + b_y= -3.25$.

Answer

Pinhole Camera

We want to take a photo from inside a house looking out a window using a pinhole camera. The camera is mounted on a tripod that maintains its view vector in the same orientation as the x-axis. The coordinates of the window’s corners are $(5.0, 0.8, -2.0), (5.0, 0.8, -3.6), (5.0, 3.8, -2.0), (5.0, 3.8, -3.6)$, and the origin is inside the house. Given that the camera has fixed focal length and aspect ratio equals 1, where would you place the camera so that it covers the entire window and, at the same time, only captures the minimum amount of wall (or anything but the window)?

Write your answer in the form $(x, y, z)$. For example, a possible answer could be:

$3.3, 0.3, -3.6$

Check your answer with the following information: adding the three coordinates of the answer adds to zero, i.e., $x+y+z=0$

Perspective

A square in 3D space is projected onto the screen using a linear perspective transformation. The 2D coordinates of the square after projection are $(−1.0, −1.0), (1.0, −1.0), (−0.6, 1.0), (0.6, 1.0)$.

Considering the vanishing lines generated by the projected square, what are the 2D coordinates of the vanishing point $P$ that is not at infinity?

• $(0.0, 2.5)$
• $(1.0, 5.0)$
• $(0.0, 3.75)$
• $(0.0, 4.0)$

Description

Develop an image filter that blurs the input image with a weighted average filter. Similar to the box filter from the lecture, this filter should use the pixel’s direct neighbour (3x3 neighbourhood). However, in this case, not all pixels contribute equally:

• the central pixel’s weight is zero;
• the diagonal neighbours’ weight is 1;
• other pixels’ (left, right, top, bottom) weight is 2.

Border pixels should be treated. In this case, use only the existing neighbouring pixels for the filter.

There are three test images: checkerboard, flowers and random.

Here are the reference results for these three input:

Test Data is Available

Solution

#include "library.cpp"

// You can access an image i-th pixel by: input.pixels[i][c], where c={0,1,2} (color channels)
// Note that you need to access the images with a single index, each pixel is a glm::vec3
// Likewise you can write to a pixel using the same notation

// You can query the image width and height by: image.width or image.height
// All images are in RGB format (three channels)
void filter (const Image &input, Image &output) {

}

int main() {

    // you can change the image to flowers.png for further testing
    // you can also change to random.png.
    std::string image_name = "../checkerboard.png";

    Image input (image_name);

    // for simplicity, create an ouput image as a copy of input (could have been a blank image)
    // running the program without modifications should output the same image as the input
    Image output (image_name);

     (input, output);

    output.();
}

Test Suite

Library

// Suppress warnings in third-party code.
#include "library.cpp"

// Suppress warnings in third-party code.
#include <framework/disable_all_warnings.h>

DISABLE_WARNINGS_PUSH()
#include <glm/glm.hpp>
#include <glm/vec2.hpp>
#include <glm/vec3.hpp>
#include <glm/vec4.hpp>
DISABLE_WARNINGS_POP()

#include <iostream>
#include <framework/image.h>

Description

We want to install a projector. Unfortunately, it cannot be placed perfectly aligned with the diffuse screen. Assuming the projection direction along the optical axis of the projector and the screen normal form an angle of $\frac{3}{4} \pi$ (radians), how much would one need to scale up the brightness of the projector, in order to deliver the same projected image brightness in the projection centre as a perfectly aligned projector at the same distance?

REMARK: “how much would one need to scale up the brightness of the projector?” asks about a factor that needs to be multiplied by the original brightness.

Select one option

• 1.75
• $2\pi$
• $4/\pi$
• $\sqrt{2}$
• Skip Question

Discussion with Course Staff

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LED lamps often combine several colored lamps to simulate white light. Assume S1, S2, S3 are light sources with RGB emission $\alpha_1$ (0.3,0.4,0.2), $\alpha_2$(0.2,0.1,0.2), $\alpha_3$(0.2,0.1,0.4) respectively. Calculate the values of , such that the added emission matches (1,1,1).

Write your answer in the following format: $\alpha_1, \alpha_2, \alpha_3$ (number comma number comma number)

Check your answer with the following hash: $\alpha_1^2 + 2.5\alpha_2 + \frac{\alpha_3}{2} = 7$

Appearance

Iridescence (or goniochromism) is a light-matter phenomenon where the surface appearance changes color (not only intensity) as the light and view direction change. Common examples are butterfly wings, soap bubbles and some car paints.

Consider a simplified iridescence effect that assumes the material to be diffuse, but its diffuse coefficient $k_d$ is given by $(2\alpha, \beta, 5\alpha^2\beta)$, where is the cosine between the surface normal and light direction, and is the cosine between the surface normal and the view direction.

Consider a point $p=(0,0,0)$ at the material surface with normal direction $n=(0, 1, 0)$. Further, consider a point light source at position $L=(-2,1,$ that emits white light. Compute the perceived color for a viewer at position , looking at point .

Write your answer in the form: R, G, B (number comma number comma number).

Check you answer with the following hash: $R+G+B=1.2$

During triangle rasterization you can use an Axis-Aligned-Bounding-Box (AABB) to reduce the search space; limit the number of pixels whose centers will be checked for being inside/outside the triangle.

Consider now that we want to approximate 2D circles by using a GL_TRIANGLE_FAN primitive. It is similar to GL_TRIANGLE_STRIP, but after the three first vertices, every new vertex connects to the first and last vertex. So, for a sequence of vertices $\{ v_0, v_1, v_2, v_3, v_4, v_5 \}$ the following triangles are created .

To create approximations of a circle in 2D space we use the method below. Note that we are using 2D vertices instead of 3D.

// input is center of circle and vector from center to first point
void draw (c_x, c_y, v_x, v_y) {

    float x = v_x;
    float y = v_y;
    
    float cos = 1.0f/sqrt(2.0f);
    float sin = 1.0f/sqrt(2.0f);

    glBegin(GL_TRIANGLE_FAN);
    glVertex2f ( c_x, c_y);
    glVertex2f ( c_x+x, c_y+y);

    for (int i = 0; i < 8; ++i) {
        float px = cos*x - sin*y;
        float py = cos*x + sin*y;
        glVertex2f ( c_x+px, c_y+py);        
        x = px;
        y = py;
    }
    glEnd();
}

The rasterizer is now modified to create and then rely on AABBs for these fans that approximate circles.

If we call the method with parameters

draw (3.5, 4.3, 2.0/sqrt(2.0), 2.0/sqrt(2.0)) {

what are the coordinates of the four corners of the respective AABB?

Write your answer in the form: $bl_x, bl_y, br_x, br_y, tr_x, tr_y, tl_x, tl_y$

where $bl$ is bottom-left, $br$ is bottom-right, $tr$ is top-right and $tl$ is top-left.

Check your answer with the following hash

$bl_x+br_x+tr_x+tl_x=14$

and

$bl_y+br_y+tr_y+tl_y=17.2$

Answer

Description

Create a function that draws a 3D spiral. The spiral should be created using line segments (GL_LINES).

It starts at the origin $(0,0,0)$ and spirals counterclockwise and outwards.

Start with a 2D spiral on the $xy$ plane. From the origin, the next vertex is translated by $(0.5, 0.0, 0.0)$ and rotated by $15^{\circ}$. For each consecutive vertex, we accumulate a new translation of $(0.5,0.0,0.0)$ and rotation of . Note that we and do not .

Once your 2D spiral works, make it spiral outwards (z-axis) by incrementing at each step $0.5$ units along this axis.

Your final spiral should be composed of 600 line segments.

Solution

#include "library.cpp"


void drawSpiral3d (void) {
    
    // create an arbitrary line as an example
    glBegin(GL_LINES);
        glVertex3f(-100.0, -100.0, -100.0);
        glVertex3f( 100.0,  100.0,  100.0);
    glEnd();

}

int main(int /* argc */, char** argv) {

    // output image resolution
    glm::ivec2 resolution (400);

    Window window { argv[0], resolution, OpenGLVersion::GL2, false }; 
    initializeGL (window);
    clearScreen(window);  
    loadMatrices (window);
    glm::mat4 transform ;

    
    transform = glm::(transform, glm::(, , ));
    transform = glm::(transform, glm::(), glm::(, , ));
    
    (glm::(transform));

    ();
    (window, );

    window.();
}

Test Suite

Library

// Suppress warnings in third-party code.
#include <framework/disable_all_warnings.h>

#include <glm/gtc/type_ptr.hpp>
#include <glm/gtx/transform.hpp>
#include <glm/gtc/matrix_transform.hpp>

#include <framework/opengl_includes.h>

#include <framework/image.h>
#include <framework/window.h>


#include <iostream>

void   {

     (!((GLADloadproc)glfwGetProcAddress)) {
        ();
        std::cerr <<  << std::endl;
        ();
    }

    window.();
    
    (, , window.().x, window.().y);

    (GL_DEPTH_TEST);

}

{
    
    (, , , );
    (GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
}

{
    (GL_PROJECTION);
    glm::mat4 projectionMatrix = glm::(glm::(), 
                        ()window.().x/()window.().y, , );
    (glm::(projectionMatrix));
    
    (GL_MODELVIEW);

}

{

    window.(filename, );
    window.();

     size = std::filesystem::(filename);

     (size > ) {
        std::cerr <<  << std::endl;
        std::cerr <<  << std::endl;
    }
}

Description

Your task is to draw a pyramid using OpenGL. The angle between a side of the pyramid and the floor is $45^\circ$. Each face should be symmetrical in regards to the positive or negative $x$ or $z$ axis (the center of the lower edge is on the axis). The pyramid floor is a 1x1 square composed of four triangles.

There should not be gaps or overlap between the triangles.

The faces should have the following colors:

• along z axis: yellow = (1.0, 1.0, 0.0)
• along -z axis: red = (1.0, 0.0, 0.0)
• along x axis: green = (0.0, 1.0, 0.0)
• along -x axis: blue = (0.0, 0.0, 1.0)
• floor: cyan = (0.0, 1.0, 1.0)

You cannot call glVertex in your code, but you can use the function drawTriangle that will render a triangle with the following coordinates: $(-\frac{1}{2}, 0, 0)^t, (\frac{1}{2}, 0, 0)^t, (0, \frac{1}{\sqrt{2}}, 0)^t$.

The following transformation methods are available from the library file.

glm::mat4 rotationMatrixX(float angle);
glm::mat4 rotationMatrixY(float angle);
glm::mat4 rotationMatrixZ(float angle);
glm::mat4 translationMatrix(glm::vec3 t);
glm::mat4 scaleMatrix(glm::vec3 s);

Below are some useful OpenGL commands (you can also find them in the code as comments):

// working with glm::mat4 and the transformations
glm::mat4 transform (1.0f); //creates an identity matrix
transform = translationMatrix (glm::vec(x,y,z)) * ;rotationMatrix (glm::radians(360.0f));

// operating with the GL_MODELVIEW matrix
glMultMatrixf(glm::value_ptr(transform)); //multiplies the current matrix by transform
glLoadIdentity(); // resets to identiy
glLoadMatrixf(glm::value_ptr(transform)); // loads a new matrix (transform)

// push and pop from the stack with
glPushMatrix();
glPopMatrix();

// set the current color
glColor3fv(glm::value_ptr(getColor(i))); // where i is the color index (see library code) 

// i=0 -> (1.0f, 1.0f, 0.0f); // yellow
// i=1 -> (0.0f, 1.0f, 0.0f); // green
// i=2 -> (1.0f, 0.0f, 0.0f); // red
// i=3 -> (0.0f, 0.0f, 1.0f); // blue

Solution

#include "library.cpp"


void drawPyramid (void) {

    // a glm 4x4 identity matrix
    glm::mat4 transform (1.0f);

    // to reset, load or multiply a matrix to the GL_MODELVIEW use
    // glMultMatrixf(glm::value_ptr(transform)); multiplies the current matrix by transform
    // glLoadIdentity(); // resets matrix on top of the stack to identity
    

    
    
    

    
    
    
    
    

    
    
    

    
    
    

    ();
}

{

    
    ;

    Window window { argv[], resolution, OpenGLVersion::GL2,  }; 
     (window);
    (window);  
     (window);
    ;

    
    transform = glm::(transform, glm::(, , ));
    transform = glm::(transform, glm::(), glm::(, , ));
    
    (glm::(transform));

    
    ();

    ();
    (window, );

    window.();
}