====== 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 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)`