Vector4.Barycentric Method (Vector4, Vector4, Vector4, Single, Single, Vector4)

[アーティクル]
12/10/2010

Returns a Vector4 containing the 4D Cartesian coordinates of a point specified in Barycentric (areal) coordinates relative to a 4D triangle.

Namespace: Microsoft.Xna.Framework
Assembly: Microsoft.Xna.Framework (in microsoft.xna.framework.dll)

Syntax

public static void Barycentric (
         ref Vector4 value1,
         ref Vector4 value2,
         ref Vector4 value3,
         float amount1,
         float amount2,
         out Vector4 result
)

Parameters

value1
A Vector4 containing the 4D Cartesian coordinates of vertex 1 of the triangle.
value2
A Vector4 containing the 4D Cartesian coordinates of vertex 2 of the triangle.
value3
A Vector4 containing the 4D Cartesian coordinates of vertex 3 of the triangle.
amount1
Barycentric coordinate b2, which expresses the weighting factor toward vertex 2 (specified in value2).
amount2
Barycentric coordinate b3, which expresses the weighting factor toward vertex 3 (specified in value3).
result
[OutAttribute] The 4D Cartesian coordinates of the specified point are placed in this Vector4 on exit.

Remarks

About Barycentric Coordinates

Given a triangle with vertices V1, V2, and V3, any point P on the plane of that triangle can be specified by three weighting factors b1, b2, and b3, each of which indicates how much relative influence the corresponding triangle vertex contributes to the location of the point, as specified in the following formulas.

    Px = (b1 * V1x) + (b2 * V2x) + (b3 * V3x);
    Py = (b1 * V1y) + (b2 * V2y) + (b3 * V3y);
    Pz = (b1 * V1z) + (b2 * V2z) + (b3 * V3z);
    Pw = (b1 * V1w) + (b2 * V2w) + (b3 * V3w);

Such triple weighting factors b1, b2, and b3 are called barycentric coordinates.

Barycentric coordinates express relative weights, meaning that (k * b1), (k * b2), and (k * b3) are also coordinates of the same point as b1, b2, and b3 for any positive value of k.

If a set of barycentric coordinates is normalized so that: b1 + b2 + b3 = 1, the resulting coordinates are unique for the point in question, and are known as areal coordinates. When normalized in this way, only two coordinates are needed, say b2 and b3, since b1 equals (1 − b2 − b3).

Bb195070.Barycentric(en-US,XNAGameStudio.10).gif

What Vector4 Barycentric Does

The Vector4 Barycentric method takes three vectors specifying the Cartesian coordinates of the triangle vertices, V1, V2, and V3), and two areal coordinates b2 and b3 of some point P (b2 is the amount1 argument and b3 is the amount2 argument). The b2 coordinate relates to vertex V2, and the b3coordinate relates to V3.

Barycentric then calculates the Cartesian coordinate of P as follows:

    Px = ( (1 - b2 - b3) * V1x ) + (b2 * V2x) + (b3 * V3x);
    Py = ( (1 - b2 - b3) * V1y ) + (b2 * V2y) + (b3 * V3y);
    Pz = ( (1 - b2 - b3) * V1z ) + (b2 * V2z) + (b3 * V3z);
    Pw = ( (1 - b2 - b3) * V1w ) + (b2 * V2w) + (b3 * V3w);

Thus, to calculate the 3D Cartesian coordinates of P, you would pass the coordinates of the triangle vertices to Barycentric together with the appropropriate normalized barycentric (areal) coordinates of P.

The following relationships may be useful.

If ( (amount1 <= 0) and (amount2 >= 0) and (1 − amount1 − amount2 >= 0) ), then the point is inside the triangle defined by value1, value2, and value3.
If ( (amount1 == 0) and (amount2 >= 0) and (1 − amount1 − amount2 >= 0) ), then the point is on the line defined by value1 and value3.
If ( (amount1 >= 0) and (amount2 == 0) and (1 − amount1 − amount2 >= 0) ), then the point is on the line defined by value1 and value2.
If ( (amount1 >= 0) and (amount2 >= 0) and (1 − amount1 − amount2 == 0) ), then the point is on the line defined by value2 and value3.

Barycentric coordinates are a form of general coordinates. In this context, using barycentric coordinates represents a change in coordinate systems. What holds true for Cartesian coordinates holds true for barycentric coordinates.

Platforms

Xbox 360, Windows XP SP2, Windows Vista

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