I have implemented the source code to create an orthonormal basis using the Gram-Schmidt orthogonalization method, so I will share it. (Dimension is fixed at 3)
Meaning and concrete examples of Gram-Schmidt orthogonalization method http://mathtrain.jp/gramschmidt
Gram-Schmidt orthonormalization method http://li.nu/blog/2010/07/gram-schmidt.html
package hoge.piyo;
import javax.vecmath.Vector3d;
public class LinearAlgebraUtil {
/**
*Vector a1, a2,Create an orthonormal basis for a3.
* a1, a2,a3 must be first-order independent.
** Argument check omitted
* @param a1 vector a1
* @param a2 vector a2
* @param a3 vector a3
* @return Orthonormal basis u1, u2,Vector3d array stored in u3 order
*/
public static Vector3d[] createOrthonormalBasis(
Vector3d a1, Vector3d a2, Vector3d a3) {
// u1
Vector3d u1 = createNormalizedVector(a1);
// u2
Vector3d v2 = new Vector3d(a2);
//Excludes components in the u1 direction from v2.
removeOrthogonalProjectionVector(v2, u1);
//Normalize v2
Vector3d u2 = createNormalizedVector(v2);
// u3
Vector3d v3 = new Vector3d(a3);
removeOrthogonalProjectionVector(v3, u1);
removeOrthogonalProjectionVector(v3, u2);
Vector3d u3 = createNormalizedVector(v3);
return new Vector3d[] {
u1, u2, u3
};
}
/**
*Get the unit vector of vector a.
* @param a vector a
* @return normalized vector
*/
private static Vector3d createNormalizedVector(Vector3d a) {
Vector3d ret = new Vector3d(a);
ret.normalize();
return ret;
}
/**
*Get the normal projection vector of the vector a in the v direction.
* @param a vector a
* @param u Unit vector of vector v
*/
private static Vector3d createOrthogonalProjectionVector(
Vector3d a,
Vector3d u) {
Vector3d ret = new Vector3d(u);
ret.scale( u.dot(a) );
return ret;
}
/**
*Remove the orthodox projection component of vector v from vector a
* @param a vector a
* @param u Unit vector of vector v
*/
private static void removeOrthogonalProjectionVector(
Vector3d a,
Vector3d u) {
Vector3d orthoProj = createOrthogonalProjectionVector(a, u);
a.sub(orthoProj);
}
}
package hoge.piyo;
import javax.vecmath.Vector3d;
import org.junit.Test;
public class LinearAlgebraUtilTest {
@Test
public void testCreateOrthonormalBasis() throws Exception {
//check the answer.
{
//Simple example
Vector3d[] S = new Vector3d[] {
new Vector3d(0, 0, 10),
new Vector3d(10, 0, 0),
new Vector3d(0, 20, 0)
};
Vector3d[] T = LinearAlgebraUtil.createOrthonormalBasis(S[0], S[1], S[2]);
dumpResult(S, T);
}
{
//Simple example 2
Vector3d[] S = new Vector3d[] {
new Vector3d(0, 0, 10),
new Vector3d(10, 10, 0),
new Vector3d(5, 20, 0)
};
Vector3d[] T = LinearAlgebraUtil.createOrthonormalBasis(S[0], S[1], S[2]);
dumpResult(S, T);
}
//Excerpt of calculation result sample from the result searched by orthonormal basis example
{
// http://www.math4all.in/public_html/linear%20algebra/chapter8.2.html
// 8.2.5 Examples:
Vector3d[] S = new Vector3d[] {
new Vector3d(1, 1, 1),
new Vector3d(-1, 0, -1),
new Vector3d(-1, 2, 3)
};
Vector3d[] T = LinearAlgebraUtil.createOrthonormalBasis(S[0], S[1], S[2]);
dumpResult(S, T);
}
{
// http://lyle.smu.edu/emis/8371/book/chap3/node12.html
// Examples:
Vector3d[] S = new Vector3d[] {
new Vector3d(1, 2, 2),
new Vector3d(1, 1, 0),
new Vector3d(1, -1, 1)
};
Vector3d[] T = LinearAlgebraUtil.createOrthonormalBasis(S[0], S[1], S[2]);
dumpResult(S, T);
}
{
// http://www.mcu.edu.tw/department/management/stat/ch_web/etea/linear/CH6.8%20--%206.9%20Orthonormal%20Bases%20in%20R%5En.pdf
// Example p6 - p12
// ->Since the answer here is orthogonal, it is necessary to normalize and think about it.
Vector3d[] S = new Vector3d[] {
new Vector3d(1, 0, 0),
new Vector3d(1, 2, 0),
new Vector3d(0, 0, 3)
};
Vector3d[] T = LinearAlgebraUtil.createOrthonormalBasis(S[0], S[1], S[2]);
dumpResult(S, T);
}
}
private static void dumpResult(Vector3d[] S, Vector3d[] T) {
System.out.println(String.format("S:%s", toString(S)));
System.out.println(String.format("T:%s", toString(T)));
System.out.println();
}
private static String toString(Vector3d[] vs) {
StringBuilder sb = new StringBuilder();
for (Vector3d v : vs) {
sb.append(String.format("(%f, %f, %f) ", v.x, v.y, v.z)).append(" ");
}
return sb.toString();
}
}
S:(0.000000, 0.000000, 10.000000) (10.000000, 0.000000, 0.000000) (0.000000, 20.000000, 0.000000)
T:(0.000000, 0.000000, 1.000000) (1.000000, 0.000000, 0.000000) (0.000000, 1.000000, 0.000000)
S:(0.000000, 0.000000, 10.000000) (10.000000, 10.000000, 0.000000) (5.000000, 20.000000, 0.000000)
T:(0.000000, 0.000000, 1.000000) (0.707107, 0.707107, 0.000000) (-0.707107, 0.707107, 0.000000)
S:(1.000000, 1.000000, 1.000000) (-1.000000, 0.000000, -1.000000) (-1.000000, 2.000000, 3.000000)
T:(0.577350, 0.577350, 0.577350) (-0.408248, 0.816497, -0.408248) (-0.707107, -0.000000, 0.707107)
S:(1.000000, 2.000000, 2.000000) (1.000000, 1.000000, 0.000000) (1.000000, -1.000000, 1.000000)
T:(0.333333, 0.666667, 0.666667) (0.666667, 0.333333, -0.666667) (0.666667, -0.666667, 0.333333)
S:(1.000000, 0.000000, 0.000000) (1.000000, 2.000000, 0.000000) (0.000000, 0.000000, 3.000000)
T:(1.000000, 0.000000, 0.000000) (0.000000, 1.000000, 0.000000) (0.000000, 0.000000, 1.000000)
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