The module "linear" includes three class definitions: Vector, Matrix, and Quaternion.
1.0 Vector
A frequently needed class of objects in aerospace navigation analysis are three dimensional vectors and matrices. A three dimensional vector, for example, can be used to represent the position of an object in space. If we pick a reference point anywhere in space and define three cartesian axes originating at that point, then the coordinates of a position, x,y, and z form three numbers which we can represent as the components of a vector.
In order to use the class Vector, include the file "linear.h" in your program, then you may declare an object of that class thus:
Vector r(0.,0.,0.);
The object r is a vector and it has been initialized to the origin of the coordinate system which is the point (0.,0.,0.).
Operator overloading has been used to implement a simple vector algebra. The following operations are supported:
Vector Addition and Subtraction:
Example
Vector r1(1.,0.,0.);
Vector r2(0.,1.,0.);
Vector r3;
Vector r4;
r3 = r1 + r2;
r4 = r1 - r2;
printf("r3 = %f %f %f\n",r3.x,r3.y,r3.z);
printf("r4 = %f %f %f\n",r4.x,r4.y,r4.z);
The vector r3 will contain the values (1.,1.,0.) and the vector r4 will contain the values (1.,-1.,0.).
Scalar or "Dot" Product of Two Vectors
Example
Vector r1; Vector r2; double s; s = r1 * r2;
The floating point number "s" contains the dot product of the two vectors r1 and r2.
A complete description of the Vector operations defined can be obtained from the comments within the implementation file, linear.cpp, and the header file, linear.h. Here is a complete list of the operations presently implemented:
// Operations: // : Vector a(x1,y1,z1); // : Vector b(x2,y2,z2); // : Vector c; // : double s; // a scalar value // add : c = a + b; // sub ; c = a - b; // dot ; s = a*b; // cross : c = a % b; // abs() : s = a.abs(); // multiply: c = a*s; vector times a scalar // divide : c = a/s; vector divided by a scalar // normalize a vector : c = a/a.abs();
2.0 Matrix
A three dimensional matrix is a collection of 9 numbers arranged in a square array. Here is a description of the Matrix class as defined in the module "linear.h."
// // class Matrix : This class is derived from vector and // : adds basic matrix operations // Initialization : #include "linear.h" // : Matrix A(m11,m12,m13, // m21,m22,m23, // m31,m32,m33); // Operations // ============ // Matrix times a matrix returns a matrix: // Matrix A; // Matrix B; // Matrix C; // C = A*B; // // Matrix times a vector, returns a vector // Vector v; // Vector u; // u = A*v; // // Transpose: // C = A.transpose(); // // Inverse: // C = A.inverse(); // // Determinant: // double d = A.det(); // // Sum of matrices // C = A + B; // // Dif of matrices // C = A - B;
In addition, the Matrix class implements to function q_to_mat. This method of the Matrix class operates on objects of the class Quaternion, defined in the next section.
3.0 Quaternion
A quaternion is an object consisting of four scalar numbers. If a quaternion has the property that it has a magnitude of 1, it is called a "versor." Versors are generally used to represent rotations of a rigid body. In aerospace navigation, Versors are sometimes used for this purpose, but the usual practice is to call them quaternions instead of versors, which are a special type of quaternion.
The quaternion consists of four scalar values, (s,v1,v2,v3). The magnitude of a quaternion is sqrt(s*s+v1*v1+v2*v2+v3*v3). At present, no algebra of quaternions has been implemented in the Quaternion class, but the Matrix class has a method to convert a quaternion to a matrix.
Example
Quaternion qbi; Matrix tr_inertial_to_body; tr_inertial_to_body = tr_inertial_to_body.q_to_mat(qbi);
All you will need to use the linear module are files linear.h and linear.cpp.