Using cosineAndSine to convert to WebMercator.

CHANGES.md

@@ -1,6 +1,10 @@
 Change Log
 ==========
 
+### 1.2 - 2014-10-01
+
+* Eliminated imagery artifacts at some zoom levels due to Mercator reprojection.
+
 ### 1.1 - 2014-09-02
 
 * Added a new imagery provider, `WebMapTileServiceImageryProvider`, for accessing tiles on a WMTS 1.0.0 server.

Source/Shaders/Builtin/Functions/cosineAndSine.glsl

@@ -0,0 +1,211 @@
+/**
+ * @private
+ */
+vec2 cordic(float angle)
+{
+// Scale the vector by the appropriate factor for the 24 iterations to follow.
+    vec2 vector = vec2(6.0725293500888267e-1, 0.0);
+// Iteration 1
+    float sense = (angle < 0.0) ? -1.0 : 1.0;
+ //   float factor = sense * 1.0;  // 2^-0
+    mat2 rotation = mat2(1.0, sense, -sense, 1.0);
+    vector = rotation * vector;
+    angle -= sense * 7.8539816339744828e-1;  // atan(2^-0)
+// Iteration 2
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    float factor = sense * 5.0e-1;  // 2^-1
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 4.6364760900080609e-1;  // atan(2^-1)
+// Iteration 3
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 2.5e-1;  // 2^-2
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 2.4497866312686414e-1;  // atan(2^-2)
+// Iteration 4
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 1.25e-1;  // 2^-3
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 1.2435499454676144e-1;  // atan(2^-3)
+// Iteration 5
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 6.25e-2;  // 2^-4
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 6.2418809995957350e-2;  // atan(2^-4)
+// Iteration 6
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 3.125e-2;  // 2^-5
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 3.1239833430268277e-2;  // atan(2^-5)
+// Iteration 7
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 1.5625e-2;  // 2^-6
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 1.5623728620476831e-2;  // atan(2^-6)
+// Iteration 8
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 7.8125e-3;  // 2^-7
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 7.8123410601011111e-3;  // atan(2^-7)
+// Iteration 9
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 3.90625e-3;  // 2^-8
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 3.9062301319669718e-3;  // atan(2^-8)
+// Iteration 10
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 1.953125e-3;  // 2^-9
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 1.9531225164788188e-3;  // atan(2^-9)
+// Iteration 11
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 9.765625e-4;  // 2^-10
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 9.7656218955931946e-4;  // atan(2^-10)
+// Iteration 12
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 4.8828125e-4;  // 2^-11
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 4.8828121119489829e-4;  // atan(2^-11)
+// Iteration 13
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 2.44140625e-4;  // 2^-12
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 2.4414062014936177e-4;  // atan(2^-12)
+// Iteration 14
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 1.220703125e-4;  // 2^-13
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 1.2207031189367021e-4;  // atan(2^-13)
+// Iteration 15
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 6.103515625e-5;  // 2^-14
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 6.1035156174208773e-5;  // atan(2^-14)
+// Iteration 16
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 3.0517578125e-5;  // 2^-15
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 3.0517578115526096e-5;  // atan(2^-15)
+// Iteration 17
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 1.52587890625e-5;  // 2^-16
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 1.5258789061315762e-5;  // atan(2^-16)
+// Iteration 18
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 7.62939453125e-6;  // 2^-17
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 7.6293945311019700e-6;  // atan(2^-17)
+// Iteration 19
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 3.814697265625e-6;  // 2^-18
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 3.8146972656064961e-6;  // atan(2^-18)
+// Iteration 20
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 1.9073486328125e-6;  // 2^-19
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 1.9073486328101870e-6;  // atan(2^-19)
+// Iteration 21
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 9.5367431640625e-7;  // 2^-20
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 9.5367431640596084e-7;  // atan(2^-20)
+// Iteration 22
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 4.76837158203125e-7;  // 2^-21
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 4.7683715820308884e-7;  // atan(2^-21)
+// Iteration 23
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 2.384185791015625e-7;  // 2^-22
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+    angle -= sense * 2.3841857910155797e-7;  // atan(2^-22)
+// Iteration 24
+    sense = (angle < 0.0) ? -1.0 : 1.0;
+    factor = sense * 1.1920928955078125e-7;  // 2^-23
+    rotation[0][1] = factor;
+    rotation[1][0] = -factor;
+    vector = rotation * vector;
+//    angle -= sense * 1.1920928955078068e-7;  // atan(2^-23)
+
+    return vector;
+}
+
+/**
+ * Computes the cosine and sine of the provided angle using the CORDIC algorithm.
+ *
+ * @name czm_cosineAndSine
+ * @glslFunction
+ *
+ * @param {float} angle The angle in radians.
+ *
+ * @returns {vec2} The resulting cosine of the angle (as the x coordinate) and sine of the angle (as the y coordinate).
+ *
+ * @example
+ * vec2 v = czm_cosineAndSine(czm_piOverSix);
+ * float cosine = v.x;
+ * float sine = v.y;
+ */
+vec2 czm_cosineAndSine(float angle)
+{
+    if (angle < -czm_piOverTwo || angle > czm_piOverTwo)
+    {
+        if (angle < 0.0)
+        {
+            return -cordic(angle + czm_pi);
+        }
+        else
+        {
+            return -cordic(angle - czm_pi);
+        }
+    }
+    else
+    {
+        return cordic(angle);
+    }
+}

Source/Shaders/Builtin/Functions/latitudeToWebMercatorFraction.glsl

@@ -15,7 +15,7 @@
  */ 
 float czm_latitudeToWebMercatorFraction(float latitude, float southMercatorYLow, float southMercatorYHigh, float oneOverMercatorHeight)
 {
-    float sinLatitude = sin(latitude);
+    float sinLatitude = czm_cosineAndSine(latitude).y;
     float mercatorY = 0.5 * log((1.0 + sinLatitude) / (1.0 - sinLatitude));
 
     // mercatorY - southMercatorY in simulated double precision.

Specs/Renderer/BuiltinFunctionsSpec.js

@@ -274,4 +274,18 @@ defineSuite([
             '}';
         verifyDraw(fs);
     });
+
+    it('has czm_cosineAndSine in all 4 quadrants', function() {
+        var fs =
+            'bool isBounded(float value, float min, float max) { ' +
+            '  return ((value < max) && (value > min)); ' +
+            '}' +
+            'void main() { ' +
+            '  gl_FragColor = vec4(isBounded(czm_cosineAndSine(czm_piOverFour).x, 0.707106, 0.707107) && isBounded(czm_cosineAndSine(czm_piOverFour).y, 0.707106, 0.707107), ' +
+            '                      isBounded(czm_cosineAndSine(czm_pi - czm_piOverFour).x, -0.707107, -0.707106) && isBounded(czm_cosineAndSine(czm_pi - czm_piOverFour).y, 0.707106, 0.707107), ' +
+            '                      isBounded(czm_cosineAndSine(-czm_piOverFour).x, 0.707106, 0.707107) && isBounded(czm_cosineAndSine(-czm_piOverFour).y, -0.707107, -0.707106), ' +
+            '                      isBounded(czm_cosineAndSine(-czm_pi + czm_piOverFour).x, -0.707107, -0.707106) && isBounded(czm_cosineAndSine(-czm_pi + czm_piOverFour).y, -0.707107, -0.707106)); ' +
+            '}';
+        verifyDraw(fs);
+    });
 }, 'WebGL');