How Quaternary is useful in determination of target location in 3D computer games?

Question

plz elaborate the concept of Axis of rotation and the Rotation Angle used in java3d to determine the shooting target in java3d games.
Is these two combined together (Axis of rot and Angle.rot) in vector form and called Quaternions.

loto app · Answer

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Utilisateur supprimé · Answer

The quaternions define a3d space with one additional dimention called axis of rotation.

Thus with little geometry giving Axis of rotation and degree of rotation .

Thus it help shooting an intercept in3d games.

formula for the quaternions,

q=q0+q1i+q2j+q3k= (q0, q1, q2, q3) q=q0+q where q0 is the scalar part and q is the vector part. formula for the quaternions,

i2 = j2 = k2 = ijk = −1,

Multiplication of basis elements

The identities

$i^2=j^2=k^2=ijk=-1$ ,

where i, j, and k are basis elements of H, determine all the possible products of i, j, and k.

For example right-multiplying both sides of −1 = ijk by k gives

$\begin{align} -k & = i j k k = i j (k^2) = i j (-1), \\ k & = i j. \end{align}$

All the other possible products can be determined by similar methods, resulting in

$\begin{alignat}{2} ij & = k, & \qquad ji & = -k, \\ jk & = i, & kj & = -i, \\ ki & = j, & ik & = -j, \end{alignat}$

Using the basis1, i, j, k of H makes it possible to write H as a set of quadruples:

$\mathbf{H} = \{(a, b, c, d) \mid a, b, c, d \in \mathbf{R}\}.$

Then the basis elements are:

$\begin{align}1 & = (1,0,0,0), \\ i & = (0,1,0,0), \\ j & = (0,0,1,0), \\ k & = (0,0,0,1), \end{align}$

and the formulas for addition and multiplication are:

$(a_1,\ b_1,\ c_1,\ d_1) + (a_2,\ b_2,\ c_2,\ d_2) = (a_1 + a_2,\ b_1 + b_2,\ c_1 + c_2,\ d_1 + d_2).$ $\begin{align} (a_1,\ b_1,\ c_1,\ d_1)&(a_2,\ b_2,\ c_2,\ d_2) = \\ & = (a_1a_2 - b_1b_2 - c_1c_2 - d_1d_2, \\ & {} \qquad a_1b_2 + b_1a_2 + c_1d_2 - d_1c_2, \\ & {} \qquad a_1c_2 - b_1d_2 + c_1a_2 + d_1b_2, \\ & {} \qquad a_1d_2 + b_1c_2 - c_1b_2 + d_1a_2). \end{align}$

If a quaternion is divided up into a scalar part and a vector part, i.e.

$q = (r,\ \vec{v}),\ q\in\mathbf{H},\ r\in\mathbf{R},\ \vec{v}\in\mathbf{R}^3$

then the formulas for addition and multiplication are:

$(r_1,\ \vec{v}_1) + (r_2,\ \vec{v}_2) = (r_1 + r_2,\ \vec{v}_1+\vec{v}_2)$ $(r_1,\ \vec{v}_1) (r_2,\ \vec{v}_2) = (r_1 r_2 - \vec{v}_1\cdot\vec{v}_2, r_1\vec{v}_2+r_2\vec{v}_1 + \vec{v}_1\times\vec{v}_2)$

where "·" is the dot product and "×" is the cross product.

the conjugation of a quaternion can be expressed entirely with multiplication and addition:

$q^* = - \frac12 (q + iqi + jqj + kqk).$

Conjugation can be used to extract the scalar and vector parts of a quaternion. The scalar part of p is (p + p*)/2, and the vector part of p is (p − p*)/2.

The square root of the product of a quaternion with its conjugate is called its norm and is denoted ||q|| (Hamilton called this quantity the tensor of q, but this conflicts with modern meaning of "tensor"). In formula, this is expressed as follows:

$\lVert q \rVert = \sqrt{qq^*} = \sqrt{q^*q} = \sqrt{a^2 + b^2 + c^2 + d^2}$

This is always a non-negative real number, and it is the same as the Euclidean norm on H considered as the vector space R4. Multiplying a quaternion by a real number scales its norm by the absolute value of the number. That is, if α is real, then

$\lVert\alpha q\rVert = |\alpha|\lVert q\rVert.$

This is a special case of the fact that the norm is multiplicative, meaning that

$\lVert pq \rVert = \lVert p \rVert\lVert q \rVert.$

for any two quaternions p and q. Multiplicativity is a consequence of the formula for the conjugate of a product. Alternatively it follows from the identity

$\det \Bigl(\begin{array}{cc} a+ib & id+c \\ id-c & a-ib \end{array}\Bigr) = a^2 + b^2 + c^2 + d^2,$

(where i denotes the usual imaginary unit) and hence from the multiplicative property of determinants of square matrices.

This norm makes it possible to define the distance d(p, q) between p and q as the norm of their difference:

$d(p, q) = \lVert p - q \rVert.$ This makes H into a metric space. Addition and multiplication are continuous in the metric topology

A unit quaternion is a quaternion of norm one. Dividing a non-zero quaternion q by its norm produces a unit quaternion Uq called the versor of q:

$\mathbf{U}q = \frac{q}{\lVert q\rVert}.$

Every quaternion has a polar decomposition q = ||q|| Uq.

Using conjugation and the norm makes it possible to define the reciprocal of a quaternion. The product of a quaternion with its reciprocal should equal1, and the considerations above imply that the product of q and q*/||q||2 (in either order) is1. So the reciprocal of q is defined to be

$q^{-1} = \frac{q^*}{\lVert q\rVert^2}.$ Thus a quaternions is a3d vector with addition of one additional dimension to facilitate the Axis og rotation.

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How Quaternary is useful in determination of target location in 3D computer games?

Multiplication of basis elements

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