In linear algebraa rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix. The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system y counterclockwise from x by pre-multiplication R on the left.

If any one of these is changed such as rotating axes instead of vectors, a passive transformationthen the inverse of the example matrix should be used, which coincides with its transpose. Since matrix multiplication has no effect on the zero vector the coordinates of the originrotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometryphysicsand computer graphics.

These combine proper rotations with reflections which invert orientation. In other cases, where reflections are not being considered, the label proper may be dropped. The latter convention is followed in this article. Rotation matrices are square matriceswith real entries. This rotates column vectors by means of the following matrix multiplication. Thus the clockwise rotation matrix is found as. The two-dimensional case is the only non-trivial i.

Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphicswhich often have the origin in the top left corner and the y -axis down the screen or page. See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix. Then according to Euler's formulaany. A basic rotation also called elemental rotation is a rotation about one of the axes of a coordinate system.

The same matrices can also represent a clockwise rotation of the axes. R zfor instance, would rotate toward the y -axis a vector aligned with the x -axisas can easily be checked by operating with R z on the vector 1,0,0 :. This is similar to the rotation produced by the above-mentioned two-dimensional rotation matrix.

See below for alternative conventions which may apparently or actually invert the sense of the rotation produced by these matrices. Other rotation matrices can be obtained from these three using matrix multiplication. For example, the product. Similarly, the product. These matrices produce the desired effect only if they are used to premultiply column vectorsand since in general matrix multiplication is not commutative only if they are applied in the specified order see Ambiguities for more details. Every rotation in three dimensions is defined by its axis a vector along this axis is unchanged by the rotationand its angle — the amount of rotation about that axis Euler rotation theorem.

There are several methods to compute the axis and angle from a rotation matrix see also axis—angle representation.

Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix.Finding the volume of a solid revolution is a method of calculating the volume of a 3D object formed by a rotated area of a 2D space. Finding the volume is much like finding the areabut with an added component of rotating the area around a line of symmetry - usually the x or y axis. Find the area of the definite integral.

Now, let's rotate this area degrees around the x axis. We will have a 3D solid that looks like this:. To find this volume, we could take vertical slices of the solid each dx wide and f x tall and add them up.

This is quite tedious, but thankfully we have calculus! Since the integrated area is being rotated around the axis under the curve, we can use disk integration to find the volume.

Since the area is rotated full circle, we can use the formula for area of a cylinder to find our volume. Volume of a cylinder. We can merge the formula for volume of a cylinder and our definite integral to find the volume of our solid. The radius for our cylinder would be the function f x and the height of our cylinder would be the distance of each disk: dx.

Since our function is linear and the radius is changing at a constant rate, it is easy to check this by plugging in values to the formula for volume of a cone. The answers are the same. Since our function was linear and shaped like a cone when rotated around the x axis, it was okay to use the volume formula for a cone.

Many of the volumes we will be working with are not shaped like cone, so we cannot simply substitute values in the formula. While algebra can take care of the nice straight lines, calculus takes care of the not-so-nice curves. The first rotated solid was integrated in terms x to find the area and rotated around the x axis. Similarly, this solid is also integrated in terms of x for the area, but it is now rotated around the y axis.

Volume of the Cylinder - Volume of the Cone.The formulas we use to find surface area of revolution are different depending on the form of the original function and the axis of rotation.

I create online courses to help you rock your math class. Read more. When the function is in the form??? Find the area of the surface generated by rotating the function about the??? Since the equation is in the form???

Using u-substitution and setting??? The surface area obtained by rotating??? Surface area of revolution around the x-axis and y-axis.

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I'm krista. Take the course Want to learn more about Calculus 2? I have a step-by-step course for that. Learn More. Finding surface area of the rotation around the x-axis over an interval Example Find the area of the surface generated by rotating the function about the???

Plugging these values back into the integral, we get???

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Get access to the complete Calculus 2 course. Get started. Learn math Krista King May 28, mathlearn onlineonline courseonline mathcalculus 2calculus iicalc 2calc iiintegralsintegrationapplications of integralsapplications of integrationintegral applicationsintegration applicationssurface arearevolutionsurface area of revolutionsurface area generatedx-axisy-axisrotation aboutrotation around.

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Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Emilio Novati Explained very well about the rotation. But I spent some time draw this picture, so I'll post it. You misunderstood the rotation matrix. You should rotate the vector counterclockwisely, instead of rotate the coordinate system.

It seems that you are confusing active and passive rotations. I believe that you are viewing the problem backwards. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 3 years, 10 months ago. Active 3 years, 10 months ago. Viewed 5k times. Okay, it seems like I should rotate vector instead of coordinate system. Then, How can you explain rotation about z axis?

## Surface area of revolution around the x-axis and y-axis

A rotation of axes in more than two dimensions is defined similarly. Coordinate systems are essential for studying the equations of curves using the methods of analytic geometry. To use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of ellipses and hyperbolasthe foci are usually located on one of the axes and are situated symmetrically with respect to the origin.

If the curve hyperbola, parabolaellipse, etc. The process of making this change is called a transformation of coordinates.

The solutions to many problems can be simplified by rotating the coordinate axes to obtain new axes through the same origin.

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Using trigonometric functionswe have. Substituting equations 1 and 2 into equations 3 and 4we obtain. Equations 5 and 6 can be represented in matrix form as. Through a change of coordinates a rotation of axes and a translation of axesequation 9 can be put into a standard formwhich is usually easier to work with.

It is always possible to rotate the coordinates in such a way that in the new system there is no x'y' term. Substituting equations 7 and 8 into equation 9we obtain. When a problem arises with BD and E all different from zero, they can be eliminated by performing in succession a rotation eliminating B and a translation eliminating the D and E terms. The conic section is:. The z coordinate of each point is unchanged and the x and y coordinates transform as above. The old coordinates xyz of a point Q are related to its new coordinates x'y'z' by.

Main article: Conic section. Categories : Functions and mappings Euclidean geometry Linear algebra Transformation function. Hidden categories: Harv and Sfn template errors Articles with short description.

Namespaces Article Talk. Views Read Edit View history. By using this site, you agree to the Terms of Use and Privacy Policy.Rotations can be represented by orthogonal matrices there is an equivalence with quaternion multiplication as described here. First rotation about z axis, assume a rotation of 'a' in an anticlockwise direction, this can be represented by a vector in the positive z direction out of the page. For an alterative we to think about using a matrix to represent rotation see basis vectors here.

The conventions used are explained here and the relationship to other standards here. This general rotation can be represented by a combination of the above rotations about Rz, Ry and Rx. As explained below the order of these rotations is important, there are different conventions for which order is assumed, as explained here. This code checks that the input matrix is a pure rotation matrix and does not contain any scaling factor or reflection for example. Successive rotations can be calculated by multiplying together the matrices representing the individual rotations.

In the same way that the order of rotations are important, the order of matrix multiplication is important. If we want to represent rotation and translation using a single matrix we need to use a 4x4 matrix as explained here.

### Rotation matrix

You may be interested in other means to represent orientation and rotational quantities such as:. 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. But no euler angles or quaternions. Terminology and Notation. Copyright c Martin John Baker - All rights reserved - privacy policy. Rotations can be represented by orthogonal matrices there is an equivalence with quaternion multiplication as described here First rotation about z axis, assume a rotation of 'a' in an anticlockwise direction, this can be represented by a vector in the positive z direction out of the page.

Other 90 degrees steps are shown here. General rotation matrix: Rotation about a general axis is: r 00 r 01 r 02 r 10 r 11 r 12 r 20 r 21 r 22 This general rotation can be represented by a combination of the above rotations about Rz, Ry and Rx.

Rotation matrices are orthogonal as explained here. So with matrix algebra different rules apply than in the algebra of numbers.Unit quaternionsalso known as versorsprovide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock.

Compared to rotation matrices they are more compact, more numerically stableand more efficient.

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Quaternions have applications in computer graphics computer visionrobotics navigationmolecular dynamicsflight dynamics orbital mechanics of satellites  and crystallographic texture analysis. When used to represent rotation, unit quaternions are also called rotation quaternions as they represent the 3D rotation group.

When used to represent an orientation rotation relative to a reference coordinate systemthey are called orientation quaternions or attitude quaternions. This is sufficient to reproduce all of the rules of complex number arithmetic: for example:. From this all of the rules of quaternion arithmetic follow, such as the rules on multiplication of quaternion basis elements.

Using these rules, one can show that:. When quaternions are used in geometry, it is more convenient to define them as a scalar plus a vector :.

Some might find it strange to add a number to a vectoras they are objects of very different natures, or to multiply two vectors together, as this operation is usually undefined. However, if one remembers that it is a mere notation for the real and imaginary parts of a quaternion, it becomes more legitimate. We can express quaternion multiplication in the modern language of vector cross and dot products which were actually inspired by the quaternions in the first place .

Quaternion multiplication is noncommutative because of the cross product, which anti-commuteswhile scalar—scalar and scalar—vector multiplications commute. From these rules it follows immediately that see details :. The left and right multiplicative inverse or reciprocal of a nonzero quaternion is given by the conjugate-to-norm ratio see details :. Our goal is to show that. Expanding out, we have. Quaternions give a simple way to encode this axis—angle representation in four numbers, and can be used to apply the corresponding rotation to a position vectorrepresenting a point relative to the origin in R 3.

This can be done using an extension of Euler's formula :.

Rotation Matrices around the Y-axis and the X-axis, 11/10/2015

In a programmatic implementation, this is achieved by constructing a quaternion whose vector part is p and real part equals zero and then performing the quaternion multiplication. Each real quaternion is carried into itself by this operation. But for the purpose of rotations in 3-dimensional space, we ignore the real quaternions.

In this instance, q is a unit quaternion and. The scalar component of the result is necessarily zero. The square of a quaternion rotation is a rotation by twice the angle around the same axis. This can be extended to arbitrary real nallowing for smooth interpolation between spatial orientations; see Slerp. Note that quaternion multiplication is not commutative.

Thus, an arbitrary number of rotations can be composed together and then applied as a single rotation. We are therefore dealing with a conjugation by the unit quaternion.