Mel Align Y Axis to Vector

Mel Align Y Axis to Vector return to main index

Introduction Occasionally it is necessary to apply a series of rotations to a coordinate system so that one of its axes points in a specific direction. This tutorial explains how the y-axis of a coordinate system can be aligned to a direction specified by a vector.  Although the calculations shown on this page were derived from "The RenderMan Companion" by Steve Upstill (pages 142/143) I have altered them slightly because RenderMan uses a left-hand coordinate system whereas Maya uses a right hand system. An additional difference between Upstill's calculations and mine is that his code transforms the (RenderMan) camera coordinate system.  Despite these differences the method by which a coordinate system is transformed is essentially the same in both Upstill's "AimZ" function and my "AimY" procedure. If you have studied Upstills code and have wondered why his calculations work I hope the illustrations on this page makes the mathematics he uses a little clearer.

Coordinate Axes Fig 1 shows a vector with coordinates [-2,2,1]. The pale red trace lines show the vector projected onto the x-y plane, the x-z plane and the y-z plane. The easiest way of figuring out how to rotate the coordinate system so that the y-axis points in the direction of the vector is to think about the problem in reverse! How can the vector be aligned to the current y-axis. The vector in question is shown below in black - coordinates [-2, 2, 1]. Figure 1

Rotations Step 1 Figure 2 shows the vector forms the hypotenuse a right angled triangle. Step 2 Figure 3 shows the triangle can be aligned to the y-z plane by applying a suitable rotation around the z-axis.

Figure 2 - vector forms a triangle

Figure 3 - rotation around the z-axis

Step 3 Fig 3 shows that a rotation around the x-axis will align the vector to the y-axis of the coordinate system (fig 4).

Figure 4 - rotation around the x-axis

Figure 5 - vector aligned with the y-axis

Angles Figure 6 shows two angles that must be calculated in order to perform steps 2 and 3. To find the blue angle we must first calculate the length of the red trace line,xyLength, on the x-y plane ie. Figure 6 - the two principle angles xyLength = sqrt(x * x + y * y); xyLength = sqrt(-2 * -2 + 2 * 2); xyLength = 2.83; After which the angle shown in blue can be found, zAngle = acos(y / xyLength); zAngle = acos(2.0 / 2.83); zAngle = 0.785; /* acos returns angle in radians */ To find the dark gray angle the length of the vector must be found, vecLength = sqrt(x * x + y * y + z * z); vecLength = sqrt(-2 * -2 + 2 * 2 + 1 * 1); vecLength = 3.0; As with the blue angle the angle in dark gray is found from the cosine. I am referring to this angle as the xAngle because, as shown in Fig 4, this angle will be used to define the rotation around the x-axis. xAngle = acos(xyLength / vecLength); xAngle = acos(2.83 / 3.0); xAngle = 0.338; Expressed in degrees the zAngle is 45.0 and the xAngle is 19.4. Therefore, the rotations needed to orientate the y-axis in the direction of the vector requires, a rotation around the x-axis of 19.4 degrees, then, a rotation around the z-axis of -45 degrees A mel procedure that implements this technique is given in listing 1.

Listing 1 proc float[] aimY(vector $vec) { float $out[2]; float $xAngle, $zAngle, $xyLength, $vecLength; $xyLength = sqrt(($vec.x) * ($vec.x) + ($vec.y) * ($vec.y)); $vecLength = sqrt(($vec.x) * ($vec.x) + ($vec.y) * ($vec.y) + ($vec.z) * ($vec.z)); // $xyLength will be zero when $vec is pointing // along the +z or -z axis if($xyLength == 0) $zAngle = ($vec.x) > 0 ? deg_to_rad(90) : deg_to_rad(-90); else $zAngle = acos(($vec.y)/$xyLength); $xAngle = acos($xyLength/$vecLength); $xAngle = ($vec.z) > 0 ? $xAngle : -$xAngle; $out[0] = rad_to_deg($xAngle); $zAngle = ($vec.x) > 0 ? -$zAngle : $zAngle; $out[1] = rad_to_deg($zAngle); return $out; }

Checking the Script The following script uses the aimY() procedure to point a cone along a vector. Fig 7 shows that the apex of the cone does extend along the vector - the coordinates of which are indicated by the gray box. Figure 7 - a cone aligned to a direction vector vector $v = <<-2,2,1>>; float $a[] = aimY($v); // begin with an empty scene select -all; delete; // insert a cone aligned to the y-axis cone -r 0.2 -hr 17 -ax 0 1 0; // apply the rotations to the x // and z axes of the cone rotate -r $a[0] 0 0; rotate -r 0 0 $a[1]; // draw a box to indicate the coordinates of the vector polyCube -w 2 -h 2 -d 1; move -r -1 1 0.5;

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© 2002- Malcolm Kesson. All rights reserved.