Fixing trigonometry

c3cat-bottle-clip/c3cat-bottle-clip.scad

@@ -480,24 +480,50 @@ module ear(depth, thickness, side_len=30, bend_factor=0.5, stretch_factor=1.2, c
   depth = depth == undef ? 20 : depth ;
   thickness = thickness == undef ? 3 : thickness ;
 
-  echo("Generating ONE single ear",
+  // $A and $B are the end of the ear
-    depth = depth,
+  // $C is the higher end point.
-    thickness = thickness,
-    side_len = side_len,
-    bend_factor = bend_factor,
-    stretch_factor = stretch_factor,
-    chamfer = chamfer,
-    chamfer_shape = chamfer_shape,
-    details = details
-  );
-
   $A=[0, side_len/2];
   $B=[0,-side_len/2];
-  $C=[-(side_len/2/sin(120))*1.5*stretch_factor, 0];
+  // From the front, an ear looks like:
+  //
+  //       $C
+  //      /|\
+  //     / | \
+  //    /  |  \
+  //   /---+---\
+  // $B    c    $A
+  //
+  // c is the origin of the ear.
+  // $Bc$C and $Ac$C are 90deg angles.
+  //
+  // The length [$B $A] is `side_len`
+  // Thus, [$B c] == [c $A] == side_len/2
+  //
+  // Since SOH, sin(angle) = Opposed / Hypotenuse
+  // sin(angle)              = Opposed / Hypotenuse
+  // sin(angle) * Hypotenuse = Opposed
+  //              Hypotenuse = Opposed / sin(angle)
+  //
+  // Thus, the `(side_len/2/sin(120))` implies that "The opposed side of the 120 angle is `(side_len/2)` length" and is the formula to know the size of the hypotenuse.
+  // Moreover, a 120deg angle in a right rectangle does not exists: the sum of all the angles of a triangle needs to be 180deg. I think the author tried, instead, to use the 60 as an angle, since the sinus is the same.
+  //
+  // The correct formula should be `((side_len/2)*tan(60))`: since $C$B$A should a 60deg angle to have an equilateral `$A$B$C` triangle, the product of the tangent ("TOA") of this angle and the adjacent segment [$B c] will give us the size of length of the opposed segment [$C c].
+  //
+  // MOREOVER, the product of the magic number `1.5` with the divisor `sin(120)`... is actually the result of the `tan(60)`!
+  //
+  // The original formula was:
+  // $C=[-(side_len/2/sin(120))*1.5*stretch_factor, 0];
+  $C=[-((side_len/2)*tan(60))*stretch_factor, 0];
+
+  // $a, $b and $c are the distance between the different points of the triangle.
   $c=sqrt(pow($A.x-$B.x, 2)+pow($A.y-$B.y, 2));
   $b=sqrt(pow($A.x-$C.x, 2)+pow($A.y-$C.y, 2));
   $a=sqrt(pow($C.x-$B.x, 2)+pow($C.y-$B.y, 2));
+
+  // $hc is the height of $C.
   $hc=-$C.x;
+
+  // If I correctly understood, $alpha should be the angle of the $B$A$C. This should be 60deg... but is not. idk why.
   $alpha=asin($hc/$b);
   $beta=$alpha;
   $gamma=180-$alpha-$beta;