# Quaternion example in math

The quaternions are a four- dimensional extension of the two-dimensional complex numbers. They form a non-associative normed division algebra over the real numbers. Sir William Rowan Hamilton was attempting to extend the complex numbers into 3 dimensions, recognizing that complex numbers could represent a 2D plane. The type of Quaternions is usually represented as after Hamilton.

It forms an Lie Algebra over the real numbers. It is typically represented as a linear combination of unit vectors and a scalar, similar to the complex numbers:.

However, if using rectangular coordinates, the axes may be chose at random, such that they follow the right-hand rule.

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There is a monomorphism from the real numbers to the quaternions, which maintains value, and the operations of the real numbers by setting the scalar quantity to the value of the real number, and leaving the vector quantities 0. Likewise, only for quaternions where the vector quantities is 0 can a quaternion be converted to a real number.

The addition of Quaternions is similar to that of a 4-vector. First, define two quaternions where:. Knowing that each individual element of A and B are real numbers: e. Again, knowing that each individual element of A and B are real numbers, the properties of addition on the real numbers imply that addition of quaternions is also associative.

The zero quaternion, represented as. Forall there is a such that:. The realization was that commutativity would need to be dropped before a multiplication was possible to be defined.

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As complex numbers are represented with a scalar-part and a vector-part, one can be represented as a linear combination :. Real numbers are represented as either bit floating-point, bit floating point, or other decimal representations. Below are some examples of quaternions implemented over float-point numbers:. This wiki.

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This wiki All wikis. Sign In Don't have an account? Categories :. Fan Feed. Universal Conquest Wiki.Quaternions add a fourth element to the [ xyz ] values that define a vector, resulting in arbitrary 4-D vectors.

However, the following example illustrates how each element of a unit quaternion relates to an axis-angle rotation, where q represents a unit quaternion xyzwaxis is normalized, and theta is the desired counterclockwise CCW rotation around the axis. Skip to main content. Contents Exit focus mode. Quaternion Structure Microsoft.

Members Table The following table lists the members exposed by the object. Fields Field Description W Retrieves or sets the w component of the quaternion. X Retrieves or sets the x component of the quaternion. Y Retrieves or sets the y component of the quaternion. Z Retrieves or sets the z component of the quaternion. Methods Method Description Add Adds two quaternions.

BaryCentric Returns a quaternion in barycentric coordinates. Conjugate Returns the conjugate of a quaternion. Dot Returns the dot product of two quaternions.

Equals Returns a value that indicates whether the current instance is equal to a specified object. Exp Calculates the exponential. GetHashCode Returns the hash code for the current instance. Invert Conjugates and re-normalizes a quaternion. Length Returns the length of a quaternion.

LengthSq Returns the square of a quaternion's length. Ln Calculates the natural logarithm. Multiply Multiplies two quaternions. Normalize Returns the normal of a quaternion. Quaternion Initializes a new instance of the Quaternion class. RotateAxis Rotates a quaternion around an arbitrary axis. RotateMatrix Builds a quaternion from a rotation matrix.

## Quaternion

RotateYawPitchRoll Builds a quaternion with the given yaw, pitch, and roll. RotationAxis Builds a quaternion that is rotated around an arbitrary axis. RotationMatrix Builds a quaternion from a rotation matrix. RotationYawPitchRoll Builds a quaternion with the given yaw, pitch, and roll. Slerp Interpolates between two quaternions, using spherical linear interpolation.

Squad Interpolates between quaternions, using spherical quadrangle interpolation. SquadSetup Sets up control points for spherical quadrangle interpolation. Subtract Subtracts two quaternion instances.This operation will be coded in the sfrotation class see this class here. Alternatively, if we want to use scalar and vector notation for quaternions, as defined on this page then addition is:.

Alternatively, if we want to use scalar and vector notation for quaternions, as defined on this page then subtraction is:.

The multiplication rules for the imaginary operators are the same as for other numbers. When we are multiplying the imaginary operators we use the following rules:. See here for more information about the rules of rotation. In mathematical terms quaternion rotation is not necessarily commutativesome subsets of quaternions may commute, for example:.

Other subsets of quaternions anti-commute, in other words reversing the order multiplies the result by This apples when different imaginary quantities, as can be seen from the list above, so for example:. In the general case quaternions containing all these parts would be neither commutative nor anti-commutative.

In order to understand more about the effect of reversing the operands it is useful to introduce the concept of the conjugate explained here. If the quaternion is unit length normalised as when used to represent rotations:. In other words, to reverse the order, take the conjugate of each part and then take the conjugate of the whole thing.

The following table summarises these issues, it shows the type and sign for each permutation of parts of the multiplying quaternions. This table summarises the character of quaternions. We can compare it with, say, geometric algebra. Notice that squaring any linear combination of imaginary terms will give a negative real number because the leading diagonal is negative and the other terms cancel out because they anticommute. So back to the general case of multiplying any two quaternions, we can just expand out the terms and group as follows:.

Alternatively, if we want to use scalar and vector notation for quaternions, as defined on this page then multiplication is:. So instead of a divide operation we multiply by the inverse. If the quaternion is unit length normalised, as it will be if we are using quaternions to represent rotations then:.

If the quaternion is not unit length then divide the conjugate by a scalar value which is the square of the magnitude of the quaternion:. Alternatively, if we want to use scalar and vector notation for quaternions, as defined on this page then division is:.

The following calculator allows you to calculate quaternion arithmetic. Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

Terminology and Notation Specific to this page here:.This might suggest that the sample is not homogeneous but possibly its elements came from two different populations, each more or less normally distributed. In such cases, in order to understand the nature of the variable in question, you should look for a way to quantitatively identify the two sub-samples. To index Purpose (What is Correlation.

The measurement scales used should be at least interval scales, but other correlation coefficients are available to handle other types of data.

Quaternions Explained Briefly

Correlation coefficients can range from -1. The value of -1. A value of 0. The most widely-used type of correlation coefficient is Pearson r, also called linear or product- moment correlation. Simple Linear Correlation (Pearson r). Pearson correlation (hereafter called correlation), assumes that the two variables are measured on at least interval scales (see Elementary Concepts), and it determines the extent to which values of the two variables are "proportional" to each other.

The value of correlation (i. This line is called the regression line or least squares line, because it is determined such that the sum of the squared distances of all the data points from the line is the lowest possible. How to Interpret the Values of Correlations. As mentioned before, the correlation coefficient (r) represents the linear relationship between two variables.

If the correlation coefficient is squared, then the resulting value (r2, the coefficient of determination) will represent the proportion of common variation in the two variables (i. In order to evaluate the correlation between variables, it is important to know this "magnitude" or "strength" as well as the significance of the correlation.

The significance level calculated for each correlation is a primary source of information about the reliability of the correlation. As explained before (see Elementary Concepts), the significance of a correlation coefficient of a particular magnitude will change depending on the size of the sample from which it was computed.

The test of significance is based on the assumption that the distribution of the residual values (i. However, Monte Carlo studies suggest that meeting those assumptions closely is not absolutely crucial if your sample size is not very small and when the departure from normality is not very large. It is impossible to formulate precise recommendations based on those Monte- Carlo results, but many researchers follow a rule of thumb that if your sample size is 50 or more then serious biases are unlikely, and if your sample size is over 100 then you should not be concerned at all with the normality assumptions.

Outliers are atypical (by definition), infrequent observations. Because of the way in which the regression line is determined (especially the fact that it is based on minimizing not the sum of simple distances but the sum of squares of distances of data points from the line), outliers have a profound influence on the slope of the regression line and consequently on the value of the correlation coefficient.

A single outlier is capable of considerably changing the slope of the regression line and, consequently, the value of the correlation, as demonstrated in the following example. Note, that as shown on that illustration, just one outlier can be entirely responsible for a high value of the correlation that otherwise (without the outlier) would be close to zero.

Needless to say, one should never base important conclusions on the value of the correlation coefficient alone (i. Note that if the sample size is relatively small, then including or excluding specific data points that are not as clearly "outliers" as the one shown in the previous example may have a profound influence on the regression line (and the correlation coefficient).

Typically, we believe that outliers represent a random error that we would like to be able to control. Unfortunately, there is no widely accepted method to remove outliers automatically (however, see the next paragraph), thus what we are left with is to identify any outliers by examining a scatterplot of each important correlation.

Needless to say, outliers may not only artificially increase the value of a correlation coefficient, but they can also decrease the value of a "legitimate" correlation. See also Confidence Ellipse. Quantitative Approach to Outliers. Some researchers use quantitative methods to exclude outliers. In some areas of research, such "cleaning" of the data is absolutely necessary.

For example, in cognitive psychology research on reaction times, even if almost all scores in an experiment are in the range of 300-700 milliseconds, just a few "distracted reactions" of 10-15 seconds will completely change the overall picture. It should also be noted that in some rare cases, the relative frequency of outliers across a number of groups or cells of a design can be subjected to analysis and provide interpretable results.

For example, outliers could be indicative of the occurrence of a phenomenon that is qualitatively different than the typical pattern observed or expected in the sample, thus the relative frequency of outliers could provide evidence of a relative frequency of departure from the process or phenomenon that is typical for the majority of cases in a group.

Correlations in Non-homogeneous Groups. A lack of homogeneity in the sample from which a correlation was calculated can be another factor that biases the value of the correlation.Remember that if you have to use a VPN to get into Bet365, online gambling is still illegal in your location.

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### Quaternion Structure (Microsoft.DirectX)

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