TDT4195/gloom-rs/report/exercise2.md

title, author, date, lang, papersize, geometry, toc, toc-title, toc-depth, numbersections, header-includes, colorlinks, links-as-notes

title	author	date	lang	papersize	geometry	toc	toc-title	toc-depth	numbersections	header-includes	colorlinks	links-as-notes
TDT4195 Exercise 2	Øyvind Nestvold Fredrik Robertsen	\today	en-US	a4	margin=4cm	false	Table of Contents	2	true		true	true

Task 1)

a-b) Color VBO and shading coloring

This task was already implemented as part of exercise 1. But for ease of access we include the same render here.

OpenGL does a simple interpolation between the vertex colors, and assigns a color to a pixel during shading based on a weighted blend of the vertex colors based on how close the location of a pixel is to the vertices of the fragment. All weights should add up to 1, resulting in a varying mix of all the vertex colors at any given point in the fragment. In other words it does a simple linear interpolation.

Task 2)

a) Transparent triangles

Here we have drawn three partially overlapping triangles drawn in the order "red - green - blue" with red having the highest z-index (furthest back), followed by green and finally blue. All triangles are rendered with alpha = 0.4

b)

I)

When reordering the color of the triangles the color of the overlapping segment shifts toward the new color. This effect is stronger the further to the front we change the color. The reason we observe this is because transparent triangles, will let some of the color of what is behind it through. When this happens we will "see" a mix of the background colors and the color of the triangle itself. Changing one color, thus changes the hue of the overlapping region.

II)

If we change the depths of the triangles, we are not drawing our triangles front to back. Alpha blending works by mixing the color of a given pixel with whatever colors are already present behind the pixels location at drawtime. If we draw the geometry behind a given triangle before its background contents, the blending won't occour, and our overlapping region will only reflect the blending of whatever geometry was drawn in a back to front order.

Task 3)

a-c)

Given a matrix defined as

\begin{bmatrix}
a & b & 0 & c \\
d & e & 0 & f \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}

we can adjust each variable a, b, c, d, e and f individually to observe all affine transformations except pure rotations. This can be intuitively understood from linear algebra, where a matrix represents a linear transformation of the basis vectors. For example, the upper left 2x2 matrix described by a, b, d and e tells us that we transform the $\mathbf{\hat{i}}$- and $\mathbf{\hat{j}}$-vectors to [a, d]^\intercal and [b, e]^\intercal respectively. Thus can these four values alone perform shears and scaling on the $x y$-plane, since these transformations are simple linear transformations as such.

• If b and d are both zero, we will perceive a scaling effect equal to a in $x$-direction and e in $y$-direction.
• Similarily, if only one of b and d are zero, we can see a shear, as one of the coordinate system vectors are only being scaled (it is being "held in place"), while the other is linearly transformed freely.
• Translation is encoded in c and f, where they correspond to translation in $x$- and $y$-directions respectively.

If we want to perform a rotation, we have to modify all four values a, b, d and e in conjunction and in accordance with the 2D rotation matrix, otherwise we end up with a shear instead. Because we have to change these values together, it is impossible to observe rotational transformations from only changing one value at a time, starting from the identity matrix.

Task 4

b)

Keybinds for the camera controls follow the default listed specifications. They are listed below:

Camera rotation:

Tilt : UP and DOWN

Yaw : LEFT and RIGHT

Translation:

Local Forward / Backward: W and S

Local Left / Right: A and D

Global up / down: SPACE and LShift

Bonus task a)

Implemented in code. Note that we have made our translation relative to only the yaw of camera. This is simply a choice of preference from our side.

We could have implemented translation as a function of both yaw and tilt, in which case our forward vector would be:

forward = \begin{bmatrix}
-cos(tilt)*sin(yaw)\\
sin(tilt)\\
cos(tilt)*cos(yaw)
\end{bmatrix}

The right vector would also have to be changed to be a rotation in 3D space, as opposed to the current XZ plane rotation.

Finally the space and LShift controls would have to be mapped to our local y-vector as opposed to a global one.

Bonus task b)

Applying some perspective to our scene, we see that the smooth interolation qualifier yields a color interpolation which is corrected in terms of perspective. In the below image you can clearly see how the colors are shifted between the smooth and noperspective interpolation models: