And rotating the point \((0,0,1)\) 90 degrees about the \(Y\) axis would result in the point \((1,0,0)\).

\[\begin{array}{ccl} v’ & = & \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \\ \\ & = & \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \end{array}\]

\(\cos{90}\) should be zero?

]]>For example:

\[ \displaystyle\pi(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} J(\sqrt[n]{x}) \]

Should produce:

\[ \displaystyle\pi(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} J(\sqrt[n]{x}) \]

]]>For example,

Einstein’s theory of special relativity is \(E=mc^2\).

And the gaussian integral is:

\[ \int_{-\infty}^\infty e^{-x^2}dx = \sqrt{\pi} \]

Should produce:

Einstein’s theory of special relativity is \(E=mc^2\).

And the gaussian integral is:

\[ \int_{-\infty}^\infty e^{-x^2}dx = \sqrt{\pi} \]

]]>You are right that the dot product between the two pure quaternions (\(p\) and \(p^{\prime}\) is 60° but in the example, \(p\) is rotated 90° about the circle that is formed by the quaternion \(q\). The dot product of the quaternions is similar to that of vectors, it measures the angle formed between the two quaternions. In this case, its the linear angle between the two vectors \(\mathbf{p}\) and \(\mathbf{p^{\prime}}\) but the the rotation is measured about the circle that is swept out by the rotation. Perhaps an image will help illustrate this.

The image visualizes the rotation as observed from looking down the \(k\) (\(z\)) axis. In this image we see that the angle between \(\mathbf{p}\) and \(\mathbf{p^{\prime}}\) is 60° but the semicircle swept out by the rotation is 90°. The next image shows it from a different angle.

This image shows the exact same rotation but now we are looking directly at the axis of rotation. Now it can easily be observed that \(p\) has been rotated exactly 90° about the quaternion axis.

So the quaternion dot product does not measure the amount of rotation that is applied, but just the angle between the vector parts of the two quaternions.

Keep in mind that if the point being rotated is very close to the axis of rotation, the circle swept by the rotation will be very small. In this case, the dot product between the original point and the rotated point may be very small (almost 0), but the amount the point was rotated could be ±180°

I hope this answers your question.

]]>The **CMakeSettings.json** file is used to specify the command-line arguments that are passed to the cmake executable when generating the solution and project files for your project. When not using Visual Studio, you can specify these command-line arguments in a batch or shell script.

See https://cmake.org/cmake/help/latest/manual/cmake.1.html for more information on the command-line arguments supported by CMake.

]]>Thank you very much! ]]>

Thanks for your explanation, this is a good complement to Microsoft explanation.

I want to start a github project in OpenGL/Vulkan C++ but sadly I am a not a real developer, I am a Java Developper ^^ so I have not a lot of experience in C++ environnement.

After several research during this week, I go to an Visual Studio Cmake environnement on Windows to integrate later and easier Vulkan sdk but I didn’t want all the visual studio stuff like .sln… on my github so their last update on Visual Studio to integrate cmake project is really interesting.

I want to share with Linux Developer and don’t let them get all the external dependencies by themselves ( like glfw3 or glm) and when I saw the CMakeSettings.json that was completely what I wanted but if I well understand it’s only for Visual Studio.

Do you know something equivalent to CMakeSettings.json to customize build for Cmake for an Linux or Mac context?

Best Regards,

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