notating theoretical grade of a 3D triangle surface on a 2d plan using a vector (represented as arrow and percentage grade) along the midpoint of the higher edge of the triangle to the lowest point on the triangle
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example triangle
position c
name x y z |\
point a 10 10 2.5 | \ ^
point b 20 10 1.5 | \ x
point c 10 20 0.5 a---b z y >
example rise
((2.5+1.5)/2)-0.5
= 1.5
example run
√(([10] - (([10] + [20])/2)² + (([10] - ([10] + [10]))/2)²))
= √(25 + 25)
= 7.07
rise / run
0.212
grade = 21.2%
theoretical formula
(c[z]- (a[z]+b[z])/2) / √((c[x] - ((a[x] + b[x])/2)² + ((c[y] - (a[y] + b[y]))/2)²)) * 100
formulas referenced
1d midpoint formula (a[z]+b[z])/2
2d midpoint formula (a[x]-a[y])+b[x]-b[y])/2
1d distance formula a[z]₂ - a[z]₁
2d distance formula √((a[x] - b[x])² + (a[y] - b[y])²)
3d distance formula √((a[x] - b[x])² + (a[y] - b[y])² + (a[z] - b[z])²)
highest z value point = point a
middle z value point = point b
lowest z value point = point c
highest z edge of triangle = line between a[z] & b[z] note: 2 highest z points in triangle, exclude lowest point
1D midpoint of highest edge = (a[z]+b[z])/2
2D midpoint between b & c (x , y) = (a[x] + b[x])/2 , (a[y] + b[y])/2
run = distance between c(x,y) and midpoint between b & c (x , y) = √((c[x] - midpoint between b & c[x])² + (c[y] - midpoint between b & c[y])²)
rise = (c[z]- 1D midpoint of highest edge[z])/2
summary:
grade = rise over run
also calculated as:
grade =
((c[z]- 1D midpoint of highest edge[z])/2) / distance between c(x,y) and midpoint between b & c (x , y) = √((c[x] - midpoint between b & c[x])² + (c[y] - midpoint between b & c[y])²)
also calculated as:
(c[z]- (a[z]+b[z])/2) / √((c[x] - ((a[x] + b[x])/2)² + ((c[y] - (a[y] + b[y]))/2)²))
as a percentage:
(c[z]- (a[z]+b[z])/2) / √((c[x] - ((a[x] + b[x])/2)² + ((c[y] - (a[y] + b[y]))/2)²)) * 100
vector = line from 1D midpoint of highest edge to 1D lowest point on triangle
alternative methods:
using a 3D distance formula along triangle edges from highest corner to lowest corner to automate labelling grades on a 2D plan, would result in vectors along the triangle edges
curving a line from midpoint of highest edge to lowest corner, ensuring 50% of the area of the triangle remains on each side of the curved line, would result in smooth curve vectors along the triangle surfaces