# Geometry tools

## Lines and distances

You can find the midpoint between two points using `midpoint`

.

The following code places a small pentagon (using `ngon`

) at the midpoint of each side of a larger pentagon:

```
sethue("red")
ngon(O, 100, 5, 0, action = :stroke)
sethue("darkgreen")
p5 = ngon(O, 100, 5, 0, vertices=true)
for i in eachindex(p5)
pt1 = p5[mod1(i, 5)]
pt2 = p5[mod1(i + 1, 5)]
ngon(midpoint(pt1, pt2), 20, 5, 0, action = :fill)
end
```

A more general function, `between`

, finds for a value `x`

between 0 and 1 the corresponding point on a line defined by two points. So `midpoint(p1, p2)`

and `between(p1, p2, 0.5)`

should return the same point.

```
sethue("red")
p1 = Point(-150, 0)
p2 = Point(150, 40)
line(p1, p2)
strokepath()
for i in -0.5:0.1:1.5
randomhue()
circle(between(p1, p2, i), 5, action = :fill)
end
```

Values less than 0.0 and greater than 1.0 appear to work well too, placing the point on the line if extended.

`center3pts`

finds the radius and center point of a circle passing through three points which you can then use with functions such as `circle`

or `arc2r`

.

`perpendicular`

finds the foot of a perpendicular line which lies on a line through two points perpendicular to a another point.

```
A, B = Point(-150, 0), Point(150, 50)
sethue("grey50")
fontsize(18)
setline(4)
line(A, B, action = :stroke)
# point perpendicular to line
C = Point(-50, -80)
D = perpendicular(A, B, C)
# line perpendicular to line
E, F = perpendicular(A, B)
# point perpendicular to extended line
G = Point(230, -200)
H = perpendicular(A, B, G)
sethue("grey50")
label.(string.(["A", "B", "C", "D", "E", "F", "G", "H"]),
:ne, offset=10,
(A, B, C, D, E, F, G, H))
sethue("red")
arrow(C, D)
sethue("green")
arrow(E, F)
sethue("orange")
arrow(G, H)
sethue("purple")
circle.([A, B, C, D, E, F, G, H], 4, action = :fill)
```

## Points and arcs

Use `isarcclockwise(c, p1, p2)`

to check whether an arc centered at `c`

running from `p1`

to `p2`

is clockwise.

The `pointinverse`

function finds the inverse of a point relative to a reference circle (centerpoint and radius). In the following example, each vertex on the shape inside the circle is linked by an arrow to its inverse outside the circle.

```
d = @drawsvg begin
radius = 60
circle(O, radius + 20, action = :stroke)
points = polycross(O, radius, 7, vertices=true)
poly(points, action = :stroke, close=true)
antipoints = last.(pointinverse.(points, O, radius+20))
for (n, pt) in enumerate(zip(points, antipoints))
sethue(HSB(length(points) * n, 0.8, 0.8))
@. circle(pt, distance(O, pt)/6, action = :fill)
sethue("black")
arrow(pt...)
end
end 800 500
```

Use `anglethreepoints`

to find the angle formed by two lines connecting three points:

```
function showangle(pt1, pt2, pt3)
θ = anglethreepoints(pt1, pt2, pt3)
label(string(round(rad2deg(θ), digits=2), "°"), :w, pt2)
newpath()
carc(pt2, 50, 0, -θ)
strokepath()
end
let
background("grey20")
sethue("white")
fontsize(12)
tiles = Tiler(800, 800, 4, 4)
for (pos, n) in tiles
@layer begin
translate(pos)
pg = [polar(50, 0), O, polar(50, n * -2π/16)]
poly(pg, action = :stroke)
for n in 1:3
pt1 = pg[1]
pt2 = pg[2]
pt3 = pg[3]
showangle(pt1, pt2, pt3)
end
end
end
end
```

Other functions that help with geometry include:

`distance`

distance between two points`getnearestpointonline`

drop perpendicular`pointlinedistance`

distance of point from line between two points`slope`

angle of line between two points`perpendicular`

find perpendicular`dotproduct`

scalar dot product of two points`@polar`

convert radius and angle to point`polar`

convert radius and angle to point`ispointonline`

true if point lies on line`ispointonpoly`

true if point lies on any edge of polygon`isarcclockwise`

true if arc is clockwise`pointinverse`

inverse of point with respect to circle`anglethreepoints`

angle formed by two lines defined by three points`determinant3`

find determinant of matrix of 3 points`rotatepoint`

rotate point around another by angle

## Triangle centers

To find the center of a triangle, use one of:

`trianglecircumcenter`

center of circumcircle/intersection of the perpendicular bisectors.`triangleincenter`

intersection of the interior angle bisectors`trianglecenter`

centroid`triangleorthocenter`

intersection of the altitudes

```
▲ = Point[Point(-100.0, 0.0), Point(110.0, 30.0), Point(65.0, 90.0)]
@layer begin
sethue("red")
setline(2)
poly(▲, :stroke, close=true)
end
# circumcenter
circle(▲..., action = :stroke)
cp = trianglecircumcenter(▲...)
circle(cp, 2, action = :fill)
label("circumcenter", :N, cp)
# incenter
cp = triangleincenter(▲...)
circle(cp, 2, action = :fill)
pt1 = getnearestpointonline(▲[1], ▲[2], cp)
@layer begin
sethue("black")
circle(cp, distance(cp, pt1), action = :stroke)
label("incenter", :S, cp)
end
# center
cp = trianglecenter(▲...)
circle(cp, 2, action = :fill)
label("center", :w, cp)
# orthocenter
cp = triangleorthocenter(▲...)
circle(cp, 2, action = :fill)
label("orthocenter", :e, cp)
```

## Intersections

`intersectionlines`

finds the intersection of two lines.

```
sethue("black")
P1, P2, P3, P4 = ngon(O, 100, 5, vertices=true)
label.(["P1", "P2", "P3", "P4"], :N, [P1, P2, P3, P4])
line(P1, P2, action = :stroke)
line(P4, P3, action = :stroke)
flag, ip = intersectionlines(P1, P2, P4, P3)
if flag
circle(ip, 5, action = :fill)
end
```

`intersectionlinecircle`

finds the intersection of a line and a circle. There can be 0, 1, or 2 intersection points.

```
l1 = Point(-100.0, -75.0)
l2 = Point(300.0, 100.0)
rad = 100
cpoint = Point(0, 0)
line(l1, l2, action = :stroke)
circle(cpoint, rad, action = :stroke)
nints, ip1, ip2 = intersectionlinecircle(l1, l2, cpoint, rad)
sethue("black")
if nints == 2
circle(ip1, 8, action = :stroke)
circle(ip2, 8, action = :stroke)
end
```

`intersection2circles`

finds the area of the intersection of two circles, and `intersectioncirclecircle`

finds the points where they cross.

This example shows the areas of two circles, and the area of their intersection.

```
c1 = (O, 150)
c2 = (O + (100, 0), 150)
circle(c1... , action = :stroke)
circle(c2... , action = :stroke)
sethue("purple")
circle(c1... , action = :clip)
circle(c2... , action = :fill)
clipreset()
sethue("black")
text(string(150^2 * π |> round), c1[1] - (125, 0))
text(string(150^2 * π |> round), c2[1] + (100, 0))
sethue("white")
text(string(intersection2circles(c1..., c2...) |> round),
midpoint(c1[1], c2[1]), halign=:center)
sethue("red")
flag, C, D = intersectioncirclecircle(c1..., c2...)
if flag
circle.([C, D], 5, action = :fill)
end
```

## Bounding boxes

The `BoundingBox`

type allows you to use rectangular extents to organize and interact with the 2D drawing area. A `BoundingBox`

holds two points, the opposite corners of a bounding box.

You can make a BoundingBox from:

- the current drawing
- two points
- a text string
- an existing polygon
- a stored path
- a table or table cell

and by modifying an existing bounding box, or using the results of functions such as `circle()`

or `box()`

.

`BoundingBox`

without arguments defines an extent that encloses the drawing (assuming that the origin is at the center of the drawing—see `origin`

). Use `centered=false`

if the drawing origin is still at the top left corner.

This example draws circles at three points: at two of the drawing's corners and at the midway point between them:

```
origin()
bb = BoundingBox()
setline(10)
sethue("orange")
circle(bb[1], 150, action = :stroke) # first corner
circle(bb[2], 150, action = :stroke) # second corner
circle(midpoint(bb...), 150, action = :stroke) # midpoint
sethue("blue")
circle.([bb[1], midpoint(bb[1:2]), bb[2]], 130, action = :fill)
sethue("red")
circle.([first(bb), midpoint(bb...), last(bb)], 100, action = :fill)
```

You can make a bounding box from a polygon:

```
p = star(O, 100, 5, 0.1, π/3.3, vertices=true)
sethue("antiquewhite")
box(BoundingBox(p), action = :fill)
sethue("black")
poly(p, action = :stroke, close=true)
```

The resulting bounding boxes can be passed to `box`

or `poly`

to be drawn.

To convert a bounding box `b`

into a box, use `box(b, vertices=true)`

or `convert(Vector{Point}, BoundingBox())`

.

To obtain the coordinates of the corners or key points on the bounding box, use the functions with names combining `box`

and `top|middle|bottom`

and `left|center|right`

. So `boxtopleft(bbox)`

finds the top left corner of the bounding box `bbox`

.

You can also do some arithmetic on bounding boxes. In the next example, the bounding box is created from the text "good afternoon". The bounding box is filled with purple, then increased by 40 units on all sides (blue), also scaled by 1.3 (green), and also shifted by `(0, 100)`

(orange).

```
translate(-130,0)
fontsize(40)
str = "good afternoon"
sethue("purple")
box(BoundingBox(str), action = :fill)
sethue("white")
text(str)
sethue("blue")
modbox = BoundingBox(str) + 40 # add 40 units to all sides
poly(modbox, action = :stroke, close=true)
sethue("green")
modbox = BoundingBox(str) * 1.3
poly(modbox, action = :stroke, close=true)
sethue("orange")
modbox = BoundingBox(str) + (0, 100)
poly(modbox, action = :fill, close=true)
```

You can find the union and intersection of BoundingBoxes, and also find whether a point lies inside one. The following code creates, shrinks, and shifts two bounding boxes (colored yellow and pink), and then draws: their union (a bounding box that includes both), in black outline; and their intersection (a bounding box of their common areas), in red. Then some random points are created (you can pass a bounding box to `rand()`

to get a random point inside the box) and drawn differently depending on whether they're inside the intersection or outside.

```
origin()
setopacity(0.75)
setline(8)
bbox1 = BoundingBox()/2 - (50, 30)
sethue("yellow")
box(bbox1, action = :fill)
bbox2 = BoundingBox()/2 + (50, 30)
sethue("pink")
box(bbox2, action = :fill)
sethue("black")
box(bbox1 + bbox2, action = :stroke)
sethue("red")
bothboxes = intersectboundingboxes(bbox1, bbox2)
box(bothboxes, action = :fill)
for i in 1:500
pt = rand(bbox1 + bbox2)
if isinside(pt, bothboxes)
sethue("white")
circle(pt, 3, action = :fill)
else
sethue("black")
circle(pt, 2, action = :fill)
end
end
```

To find out where a line starting at the center of a bounding box passing through a point crosses or would cross the edges of the box, use `pointcrossesboundingbox`

.

```
bx = BoundingBox(box(O, 200, 200))
setline(1)
box(bx, action = :stroke)
for i in 1:10
pt = randompoint((1.5bx)...)
pt2 = pointcrossesboundingbox(pt, bx)
sethue("grey50")
arrow(O, pt)
sethue("red")
circle(pt2, 3, action = :stroke)
end
```

## Random points

You can use `randompointarray`

to create an array of randomly placed points.

The `randompointarray(boundingbox, distance)`

method fills the boundingbox with random points up to `distance`

units apart using a Poisson Disk sampling method.

```
background("black")
b = blend(
boxtopleft(BoundingBox()),
boxbottomright(BoundingBox()),
"red",
"green")
addstop(b, 0.3, "orange")
addstop(b, 0.4, "magenta")
addstop(b, 0.5, "cyan")
addstop(b, 0.7, "yellow")
setblend(b)
for pt in randompointarray(BoundingBox() * 0.9, 15)
d = rescale(distance(pt, O), 0, sqrt(800 * 500), 1, 0)
circle(pt, 1 + 7d, action = :fill)
end
```

The `randompointarray(point1, point2, n)`

method generates `n`

random points in the area bounded by two points, using Julia's random number generator.

```
pt1 = Point(-300, -150)
pt2 = Point(300, 150)
sethue("purple")
map(pt -> circle(pt, 6, action = :fill), (pt1, pt2))
box(pt1, pt2, action = :stroke)
sethue("blue")
map(pt -> circle(pt, 4, action = :fill), randompointarray(pt1, pt2, 200))
```

Use `rand(BoundingBox())`

to return a single point somewhere inside a bounding box.

## Noise

For artistic graphics you might prefer noisy input values to purely random ones. Use the `noise`

function to obtain smoothly changing random values corresponding to input coordinates. The returned values wander slowly rather than jump about everywhere.

In this example, the gray value varies gradually as the `noise`

function returns values between 0 and 1 depending on the location of the two input values `pos.x`

and `pos.y`

.

The top two quadrants use a lower value for the `detail`

keyword argument, an integer ()>= 1) specifying how many "octaves" of noise you want.

The left two quadrants use a lower value for the `persistence`

keyword argument, a floating point number specifying how the amplitude diminishes for each successive level of detail. There is more fine detail when the persistence is higher, particularly when the `detail`

setting is also high.

```
tiles = Tiler(800, 400, 200, 200)
sethue("black")
for (pos, n) in tiles
freq = 0.05
pos.y < 0 ? d = 1 : d = 4
pos.x < 0 ? pers = 0.3 : pers = 1.0
ns = noise(freq * pos.x, freq * pos.y,
detail=d,
persistence=pers)
setgray(ns)
box(pos, tiles.tilewidth, tiles.tileheight, action = :fillstroke)
end
```

Use `initnoise`

to initialize the noise behaviour.