2D Graphics

Formula for a line

`y = mx + b`

Slope of a line

`m = (Delta y)/(Delta x) = (y_2 - y_1)/(x_2 - x_1)`

Y-intercept

Y-intercept - where the line intersects with the line defined by the Y-axis
The line intersects the Y-axis when `x` is `0`.

amath y = mx + b
y = m(0) + b
y = 0 + b
y = b
endamath

Point-Slope form

`y = m(x - x_1) + y_1`

Perpendicular Lines

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. For example, given the line: `y = mx + b`, the slope of a line perpendicular is: `m_1 = -1/m`

Perpendicular lines will intersect at a single point. The point of intersection is where the `x` and `y` values satisfy both equations.

Given a point on the line, find the equation of the line perpendicular to that line intersecting at that point:

Line: `y = 1/2 x + 2`
Intersection: `(2, 3)`

a``graph width=320; height=220; xmin=-8.3; xmax=8.3; ymin=-3.3; ymax=8.3; xscl=1; yscl=1; plot(1/2*x + 2); enda``graph <html> <embed class="ASCIIsvg" script=' width=320; height=220; xmin=-8.3; xmax=8.3; ymin=-3.3; ymax=8.3; xscl=1; yscl=1; plot(1/2*x + 2); '/> </html>

We can find the slope of the second line by taking the negative reciprocal of the slop of the first: `m_1 = -2/1 = -2`

Next we need to find the Y-intercept. In this case, we know the slope of the line and a point that satisfies the line. All we need to do is plug in those values and solve for b:

amath y = mx + b
3 = -2 * 2 + b
3 = -4 + b
b = 7
endamath

So, the equation of the perpendicular line is: `y = -2x + 7`

Intersection of two lines

Two lines intersect when the point `(x, y)` satisfies both equations. To determine the point of intersection of two lines:

`y_1 = 1/2 x_1 + 2`
`y_2 = -2x_2 + 7`

Given that `y_1 = y_2` we can write:

`1/2 x_1 + 2 = -2x_2 + 7`

Since `x_1 = x_2` we can write:

amath 1/2 x + 2 = -2x + 7

Now solve for x:

1/2 x + 2x + 2 = 7
5/2 x + 2 = 7
5/2 x = 5
5x = 10
x = 2 endamath

Now that we have an `x` value we can use one of the equations to find the `y`.

amath y = 1/2 x + 2
y = 1/2 * 2 + 2
y = 1 + 2
y = 3 endamath

The point of intersection of the two lines is `(2, 3)`

Find point on a line

Given a point on a line `(x, y)` a slope `m` and a distance `d` along the line from the given point, what is the `(x_1, y_1)` of the new point?

`c = cos(theta) = 1/sqrt(1 + m^2)`
`s = sin(theta) = m/sqrt(1 + m^2)`
`x_1 = x + d * c`
`y_1 = y + d * s`

Pythagorean theorem

`c^2 = a^2 + b^2` or `c = sqrt(a^2 + b^2)`

Fourth Woods Wiki

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