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--- navigation_trigonometry [2022/09/23 01:52] – [Units Conventions] jhagstrand
+++ navigation_trigonometry [2023/01/12 11:34] (current) – removed jhagstrand
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-====== Navigation Trigonometry ======
-This page provides knowledge to backup the functions in **nav.py**
-see also: https://docs.google.com/spreadsheets/d/1LO3Nu2v9EHzgbMTeB59HaXaxw9gm1k48L6JluEeYWGw/edit#gid=0
-=== route ===
-a series of lines and arcs between points on a map
-=== point ===
-can be defined relative to a center point in two ways:
-  * by cartesian coordiates: x,y
-  * by polar coordinates: theta, radius
-=== line ===
-line can be represented in two ways:
-  + by two points
-  + by starting point, heading, and length
-an arc can be represented as:
-  * two thetas, radius, direction (clockwise or counter-clockwise)
-===== Drop a Triangle =====
-Example navigation problem: **given a line AB, find the heading and length of the line**
-Solution:
-Add point C to make a right triangle ABC, such that
-  * point C is the right angle c
-  * line AB is the hypotenuse
-  * angle a at point A is parallel to the x-axis, with length dx from point C
-  * angle b at point B is parallel to the y-axis, with length dy from point C
-From the two points A,B, we know:
-  * angle c = 90 degrees
-  * dx = xB - xA, horizontal vector, length of the adjacent side
-  * dy = yB - yA, vertical vector, length of the opposite side
-  * slope of AB = dy/dx, the ratio of vertical over horizontal
-Now using the equations below we can calculate the remaining factors:
-  * length of AB = square root of (dx squared + dy squared) # pythagorean theorem
-  * angle a = arcsin(dy / AB)   # inverse trigonometry functions
-  * angle b = arccos(dx / AB)
-===== Trigonometry Functions =====
-The primary functions return the ratio of the sides, per "soh cah toa".
-  * sin(a) = opp/hyp
-  * cos(a) = adj/hyp
-  * tan(a) = opp/adj
-The inverse functions return the angle, given the ratio of the two sides.
-  * a = arcsin(opp/hyp)
-  * a = arccos(adj/hyp)
-  * a = arctan(opp/adj)
-===== Pythagorean Theorem =====
-        sq(dx) + sq(dy) = sq(hyp)
-===== Law of Sines =====
-The ratios of each side over the sine of its opposing angle are equal.
-                opp(a)/sin(a) = opp(b)/sin(b)  = opp(c)/sin(c)
-see: https://www.mathsisfun.com/algebra/trig-sine-law.html
-===== Terms =====
-^ object           ^ standard                             ^ indicates                  ^ system        ^
-| point            | (x,y)                                | place                      | cartesian     |
-| line             | (A, B)                               | place, direction, distance | cartesian     |
-| vector           | [dx,dy]                              | direction, distance        | cartesian     |
-| length           | signed float                         | distance                   | cartesian     |
-| slope            | signed float                         | vague direction            | cartesian     |
-| angle            | radians (or degrees)                 | vague direction            | polar         |
-| theta            | radians (or degrees or piscalar      | direction                  | polar         |
-| quadrant         | integer (1-4)                        | vague direction            | polar         |
-| ray              | center, theta                        | direction, place           | polar         |
-| polar point      | center, theta, radius                | place                      | polar         |
-| arc              | center, theta1, theta2, radius, wise | place, direction, distance | polar         |
-| wise             | 'cw' or 'ccw'                        | rotational direction       | polar         |
-| heading          | degrees                              | direction                  | mercator map  |
-| course           | degrees                              | direction                  | mercator map  |
-| bearing          | degrees                              | direction                  | mercator map  |
-| relative bearing | degrees                              | direction                  | mercator map  |
-        Heading - direction the vehicle is pointing
-        Course - direction the vehicle is moving, may be different from heading due to drift
-        Bearing - direction to destination or navigational aid
-        Relative bearing - angle between heading and bearing
-        angle - the angle within the right triangle, always < 90 degrees (in radians)
-        theta - the obtuse angle indicating direction within the circle (in radians)
-===== Units Conventions =====
-  radians - 2pi to a circle, oriented at 3 o'clock
-  compass degrees - 360 to a circle, oriented to 12 o'clock
-  heading, course and bearing are given in compass degrees
-  relative bearing is given in degrees
-  angle and theta are given in radians
-  an inverse function undoes the function
-  arctangent is the inverse of tangent
-  ratio = tan(angle)      # tangent takes an angle and returns a ratio
-  angle = arctan(ratio)   # arctangent takes the ratio and returns the angle
-  slope = tan(theta)      # toa, ratio of opposite to adjacent, y/x, slope
-  theta = arctan(slope)   # inverse of tangent, get the angle theta from the slope
-  slope does not distinguish between lines going up or down
-  positive slope indicates a direct relationship between x and y
-  negative slope indicates an inverse relationship between x and y
-  negative change in y means line is going down, else up
-  negative change in x means line is going left, else right
-  theta does distinguish between lines going up or down
-  theta is an obtuse angle between 0 and 2pi
-  theta between 0 and pi indicates the line is going up
-  theta between pi and 2pi indicates the line is going down
-  . -dy, -dx => +slope, down to the left , quadrant 3 ll
-  . +dy, +dx => +slope, up   to the right, quadrant 1 ur
-  . -dy, +dx => -slope, down to the right, quadrant 4 lr
-  . +dy, -dx => -slope, up   to the left , quadrant 2 ul
-  theta should always be positive
-  arctan(negative slope) returns a negative theta
-  as slope approaches vertical from the right it approaches infinity
-  as slope approaches vertical from the left it approaches -infinity
-  an angle can be expressed as radians or degrees
-  let angle refer to the angle within a right triangle, always < 90 degrees or pi/2
-  let theta refer to the obtuse angle relative to the right-side x axis
-  slope = ratio dy/dx = tan(angle)   # tan() returns a ratio
-  angle = arctan(dy/dx)              # arctan() returns an angle in radians
-  -90 degrees < angle < +90 degrees
-   -1.57 rads < angle < +1.57 rads      # pi/2 = 1.57
-if dy/dx is negative, arctan(dy/dx) is negative (in quadrants 2 and 4)
-Summary
-        slope -> angle -> theta -> heading
-  sides -> ratio -> angle -> theta -> heading
-  dx,dy -> dy/dx -> arctan() -> theta -> heading
-  sides = dx,dy
-  ratio = dy/dx
-  angle = arctan(ratio)
-  theta = thetaFromAngle(angle,dx,dy)
-  heading = headingFromTheta(theta)
-  given a line AB
-  where
-    slope is the ratio of dx,dy,  between -inf and +inf
-    angle is the angle of our line to the x-axis, between -pi/2 and +pi/2 in radians
-    theta is between +=2pi radians, oriented to horizontal axis pointing right
-    heading is between 0-360 degrees, oriented to straight up north
-    quadrant is 1,2,3, or 4
-   ----slope--->    angle-> -----theta---->  heading   quadrant
-                             rads   *pi degr
-   +inf +dy / 0dx     1.57   1.57  0.50   90        0   vertical north
-+dy / +dx     0.79   0.79  0.25   45       45   ur quadrant 1
-0dy / +dx        0   0     0       0       90   horizontal east
-     -1 +dy / -dx    -0.79   5.50  1.75  315      135   lr quadrant 4
-   -inf -dy / 0dx    -1.57   4.71  1.50  270      180   vertical south
--dy / -dx     0.79   3.93  1.25  225      225   ll quadrant 3
-0dy / -dx        0   3.14  1.00  180      270   horizontal west
-     -1 +dy / -dx    -0.79   2.35  0.75  135      315   ul quadrant 2
-===== Cartesian vs Polar Coordinates =====
-        cartesian coordinate point: x,y
-        polar coordinate point: center, theta, radius
-polar coordinates
-        as if, looking at a globe at the north pole
-        picture a globe
-        rotatated so that you are looking down on the north pole
-        the prime meridian is on the right, and the international dateline is on the left
-        that means the meridians are going counter-clockwise from 0 to 180
-        if you go counterclockwise, the meridians are -1 to -180
-        therefore,
-        theta is always positive
-        angle can be negative or positive
-        theta originates at the positive x-axis
-        angle originates at the x-axis, either negative or positive
-        angle is relative to the quadrant
-        theta is relative to the whole circle
-        to calculate
-        drawing counter-clockwise, the angle will be negative
-        draw from 40 degrees to -o
-        a point can be positioned
-        a line between two points can only be drawn one way
-        an arc between two points can be drawn two ways: cw and ccw
-        wise = cw
-        wise = ccw
-        cw = -1
-        ccw = 1
-        A, B, center, radius, wise
-        A, B, thetaA, thetaB, center, radius, wise
-        can you calculate a center from two points?
-                yes if you know the radius
-        can you calculate the radius?
-                yes if you know the center
-        it would be the point where the lengths are equal
-        and there would be two possible points
-if dx is 0, we don't know if the line is going up or down
-options
-        look at the last known heading
-        don't allow identical x values in the cones array
-                also scan left and right values for match
-===== Angle vs Theta =====
-        both are given in radians
-        theta indicates direction, angle does not
-        in fact, angle can indicate one of four thetas, depending on quadrant
-        the signs of dx,dy are required to determine quadrant
-https://socratic.org/questions/how-do-you-convert-3-4-into-polar-coordinates
-===== Heading vs Theta =====
-        heading and theta both indicate direction within a circle
-        heading is given in compass degrees, 0 to 360, clockwise
-        theta is given in radians, 0 to 2pi, counter-clockwise
-        theta is used in the matplotlib Arc() function
-        heading is used by humans
-        conversion functions:
-                theta = thetaFromHeading(heading)
-                heading = headingFromTheta(theta)
-line AB has slope, length, heading
-thetafrom, theta2
-difference between the two thetas, as a percentage of total theta for the circle 2pi
-total length of the circle = circumference = 2pir
-pct_thetadiff * circumference