2D Rendering

• Denote an image with a single array and a pixel's index can be found by width * y + x.
• The x coordinate is i % width, y is i / width.
• To get a point at a distance r and angle t, relative to point (x0, y0) $$x=x0+r\ast cos(t)$$ $$y=y0+r\ast sin(t)$$
• For drawing a simple primitive, use its derivative to get the next pixel, usually the derivative is simpler
• Ok, Fk, This stuff is sub optimal, since it kinda encourages drawing vertically, which is not optimal for the cache.

Line

• A line's equation is

$$y-y0=m\ast (x-x0)$$ or, $$x-x0=(y-y0)/m$$ where $m=(y1-y0)/(x1-x0)$

• When drawing line, a simple for (x in x0..x1) usually leaves gaps if $m>1$, and for (y in y0..y1) leaves gaps when $m<1$, use both with if cases.
• Use $y^{\prime }=m$ when $x=1$ when drawing lines to avoid division.

Circle

• A circle's equation is

$$(x-x0{)}^2+(y-y0{)}^2=r^2$$ or, $$y-y0=\sqrt{(}{r}^2-(x-x0{)}^2)$$

• Its not worth doing the derivative for a circle, just use the equation.
• A circle can be drawn by only finding 1/8th of it, and mirroring it to the other 7 parts.

Ellipse

• An ellipse's equation is

$$(x-x0{)}^2/a^2+(y-y0{)}^2/b^2=1$$ or, $$y-y0=b\ast sqrt(1-(x-x0{)}^2/a^2)$$

• Its not worth doing the derivative for a circle, just use the equation.
• An ellipse can be drawn by only finding 1/4th of it, and mirroring it to the other 3 parts.

Rotated Rectangle

• $angle=angle(\mathrm{mod}\pi /2)$
• Find the points of upper line and the corresponding point at the bottom part at the same x coordinate.
• Draw a vertical line between these two points.
• Bam, you have a rotated rectangle.