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If you want to generate the Catmull-Rom spline between two base points, you also need to know the two adjacent base points. In the example of Figure 5-35, when you want to generate the part of the curve between base points 1 and 2, you need to know the coordinates of base points 0, 1, 2, and 3. The XNA Framework already comes with 1D Catmull-Rom functionality. You can use the MathHelper.CatmullRom method to pass it any four base points, and it will calculate for you any point on the curve between the second and third points. You must also pass a fifth value, between 0 and 1, indicating which point between the second and third points you want. In this recipe, you ll extend this functionality to 3D and create a sample method that uses this to generate a spline between multiple base points.
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XNA provides 1D Catmull-Rom interpolation between single values. Because a Vector3 is nothing more than a combination of three single values, you can obtain Catmull-Rom interpolation by calling the 1D XNA method on the X, Y, and Z components of your Vector3s. The following method is the 3D extension of the default XNA method. Instead of four values, it accepts four Vector3s.
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private Vector3 CR3D(Vector3 v1, Vector3 v2, Vector3 v3, Vector3 v4, float amount) { Vector3 result = new Vector3(); result.X = MathHelper.CatmullRom(v1.X, v2.X, v3.X, v4.X, amount); result.Y = MathHelper.CatmullRom(v1.Y, v2.Y, v3.Y, v4.Y, amount); result.Z = MathHelper.CatmullRom(v1.Z, v2.Z, v3.Z, v4.Z, amount); return result; } This method will return a Vector3, called result, on the spline between v2 and v3. The amount variable allows you to specify the distance between v2, the result, and v3, where 0 will return v2, and 1 will return v3.
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Using the CR3D Method to Calculate the Spline
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Now that you have a method capable of doing 3D Catmull-Rom interpolation, let s use it to generate multiple points of a spline. Given the four base points v1, v2, v3, and v4 in 3D space, the following method returns a list of 20 points on the spline between v2 and v3, with the first point in the returned list being v2. private List<Vector3> InterpolateCR(Vector3 v1, Vector3 v2, Vector3 v3, Vector3 v4) { List<Vector3> list = new List<Vector3>(); int detail = 20; for (int i = 0; i < detail; i++) { Vector3 newPoint = CR3D(v1, v2, v3, v4, (float)i / (float)detail); list.Add(newPoint); } return list; } Increase/decrease the detail value if you want to add/reduce the number of points calculated between v2 and v3 (or better, make it an additional argument of the method).
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Note The last point of the list will be the last point on the spline from v2 to v3 not being v3 itself. This is
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because in the last call to the CR3D method, you pass in 19/20, which is not equal to 1. You can always add v3 manually, for example, by adding this line before returning: list.Add(v3);.
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Joining Multiple Parts of the Spline Together
The previous code generates a spline between the two middle points of four points. For bonus points, the following code shows how you can extend the spline:
CHAPTER 5 GETTING THE MOST OUT OF VERTICES
points.Add(new points.Add(new points.Add(new points.Add(new points.Add(new points.Add(new
Vector3(0,0,0)); Vector3(2,2,0)); Vector3(4,0,0)); Vector3(6,6,0)); Vector3(8,2,0)); Vector3(10, 0, 0));
List<Vector3> crList1 = InterpolateCR(points[0], points[1], points[2], points[3]); List<Vector3> crList2 = InterpolateCR(points[1], points[2], points[3], points[4]); List<Vector3> crList3 = InterpolateCR(points[2], points[3], points[4], points[5]); straightVertices = XNAUtils.LinesFromVector3List(points, Color.Red); crVertices1 = XNAUtils.LinesFromVector3List(crList1, Color.Green); crVertices2 = XNAUtils.LinesFromVector3List(crList2, Color.Blue); crVertices3 = XNAUtils.LinesFromVector3List(crList3, Color.Yellow); First, you define a list of seven points in 3D space. Next, you use the first four points to generate extra points between points[1] and points[2]. You shift one point and get the extra points between points[2] and points[3]. Finally, you retrieve the extra points between points[3] and points[4]. This means you end up with many extra points, all positioned on a spline starting at points[1], going through points[2] and point[3], and ending in points[4] (take a look at Figure 5-35 to get a better idea). Because you ll want to render these to have a look, you create vertices from all three generated parts of the spline. You first turn the seven base points into vertices so you can render the lines as a comparison. You can do this using this simple method: public static VertexPositionColor[] LinesFromVector3List(List<Vector3> pointList, Color color) { myVertexDeclaration = new VertexDeclaration(device, VertexPositionColor.VertexElements); VertexPositionColor[] vertices = new VertexPositionColor[pointList.Count]; int i = 0; foreach (Vector3 p in pointList) vertices[i++] = new VertexPositionColor(p, color); return vertices; } This simply turns each Vector3 from a List into a vertex and stores it together with a Color of your choice in an array. You can render lines between all of these vertices as shown in recipe 6-1:
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