MathHelper.Barycentric Method

[アーティクル]
12/10/2010

Returns the Cartesian coordinate for one axis of a point that is defined by a given triangle and two normalized barycentric (areal) coordinates.

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

Syntax

public static float Barycentric (
         float value1,
         float value2,
         float value3,
         float amount1,
         float amount2
)

Parameters

value1
The coordinate on one axis of vertex 1 of the defining triangle.
value2
The coordinate on the same axis of vertex 2 of the defining triangle.
value3
The coordinate on the same axis of vertex 3 of the defining triangle.
amount1
The normalized barycentric (areal) coordinate b2, equal to the weighting factor for vertex 2, the coordinate of which is specified in value2.
amount2
The normalized barycentric (areal) coordinate b3, equal to the weighting factor for vertex 3, the coordinate of which is specified in value3.

Return Value

Cartesian coordinate of the specified point with respect to the axis being used.

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);

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 is known to equal (1 − b2 − b3).

Bb195073.Barycentric(en-US,XNAGameStudio.10).gif

What MathHelper Barycentric Does

On an axis a (a being the x-, y-, or z-axis), the MathHelper.Barycentric method takes three triangle vertex coordinates on that axis (V1a, V2a, and V3a), 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 b3 coordinate relates to V3.

Barycentric then calculates the Cartesian coordinate of Pa as follows:

Pa = ( (1 - b2 - b3) * V1a ) + (b2 * V2a) + (b3 * V3a);

Thus, to calculate the x-axis Cartesian coordinate of P, you would pass the x-coordinates of the triangle vertices to Barycentric together with the approproiate areal coordinates of P.

Platforms

Xbox 360, Windows XP SP2, Windows Vista

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