How to calculate/get 4 points (and angle) of a rectangle by two corners

[SokoFromNZ]

Hi guys,

I'm kinda lost with my problem after hours of reading and research so I hope you can help!

I have two points (red) on a map like this:

And I like to calculate the two other points and angle in degrees of the rectangle (blue arrows) for a given aspect ratio of the rectangle:

I do not need to draw them on a map, just calculation and just in 2D Cartesian mathematics.

I've looked through all the GeometryEngine methods and couldn't find anything I can do a rotation of a Polyline for example...

It may be possible to do it with Extend and Intersections (also mentioned here) once I can draw the green rectangle... but its kinda complicated.

Anyhow: How do I get/draw the four sides of this rectangle?

Also MultipointBuilder looks promising... but can it draw a rectangle?

Finally reading through geometry help and make-measurements didn't really help. Although it mentions AngularUnits it does not say how to determine an angle of a Polyline or Segment.

Thanks for your help!

Soko

PS: Am I overthinking this to much in a big way? Can I do the calculations with simple trigonometry instead of GeoemtryEngine methods?

[JohannesLindner · Accepted solution]

Am I overthinking this to much in a big way? Can I do the calculations with simple trigonometry instead of GeoemtryEngine methods?

Probably. I don't know anything about the .Net SDK, but here is what you could do in pure Python (without  using the arcpy geometry methods):

import math

# some basic trigonometry functions
# points as coordinate tuples
# angles in radians
def distance(p1, p2):
    return math.sqrt(math.pow(p1[0] - p2[0], 2) + math.pow(p1[1] - p2[1], 2))
def angle(p1, p2):
    return math.atan2(p2[1] - p1[1], p2[0] - p1[0])
def point_from_angle_and_distance(p, a, d):
    return (p[0] + d * math.cos(a), p[1] + d * math.sin(a))
    
    
def get_rectangle(p1, p2, aspect):
    """ Outputs the vertices of the (a) rectangle defined by the two given corner points and the aspect ratio.
    
    p1, p2: Coordinate tuples of two points in a diagonal of the rectangle
    aspect: ratio of the two sites (<= 1)
    """
    # get the center of the diagonal
    a_p1p2 = angle(p1, p2)
    d_p1p2 = distance(p1, p2)
    center = point_from_angle_and_distance(p1, a_p1p2, d_p1p2 / 2)
    
    # convert the aspect ratio to angle between the diagonals
    a_diag = math.pi - 2 * math.atan(aspect)
    
    # get the other two points
    p3 = point_from_angle_and_distance(center, a_p1p2 - a_diag, d_p1p2 / 2)
    p4 = point_from_angle_and_distance(center, a_p1p2 + math.pi - a_diag, d_p1p2 / 2)
    
    # reorder and return
    return [p1, p3, p2, p4]

Now let's give that method a spin... We will read from a Multipoint layer, convert the multipoints to coordinate tuples, calculate the resulting rectangle for multiple aspect ratios, convert those rectangles into arcpy.Polygons and write them into a polygon layer.

aspect_ratios = [0.01, 0.25, 0.5, 0.75, 1]
with arcpy.da.InsertCursor("TestPolygons", ["SHAPE@", "DoubleField1"]) as i_cursor:
    with arcpy.da.SearchCursor("TestMultipoints", ["SHAPE@"]) as s_cursor:
        for shp, in s_cursor:
            p1 = (shp.firstPoint.X, shp.firstPoint.Y)
            p2 = (shp.lastPoint.X, shp.lastPoint.Y)
            for aspect in aspect_ratios:
                rect_points = get_rectangle(p1, p2, aspect)
                vertices = [arcpy.Point(p[0], p[1]) for p in rect_points]
                poly = arcpy.Polygon(arcpy.Array(vertices), spatial_reference=arcpy.SpatialReference(25833))
                i_cursor.insertRow([poly, aspect])

 Result:

So yeah, you just need basic trigonometry for this.

[JohannesLindner]

PS: This problem has two solutions, depending on which diagonal the two point describe. If you change line 26 to take the arctan of 1/aspect, you get the second solution.

Left: solution 1 -> atan(aspect), Right: solution 2 -> atan(1/aspect)

[SokoFromNZ]

Hi Johannes,

and thanks for the great explanation and phyton code. I'll give it a try to port it to C#.

About the two solutions: The two points are always lower-left, upper-right. So solution 1 is the right one for me.

One thing though I'm worried about:
This pure trigonometry solution works for sure in a "standard" coordinate system. By "standard" I mean were X and Y distances/units are the same.

But longitude and latitute distances are not the same afaik. Apparently even one unit/degree is a different distance depending where you are on the globe.

Do you think this matters?

Soko

[JohannesLindner]

Oh right. this calculation is for cartesian coordinate systems, as per your question.

I have only worked with projected coordinate systems until now, so I'm not really sure how this will play out in a geographical coordinate system. The error might be negligible over small distances, but for greater distances, it will probably become apparent.

Maybe you can get away with converting the points into a locally applicable projected coordinate system, do the calculation, and then project the resulting points into the geographic system.