To plot more than one function at once, simply replace the single function with a list of functions. But its real hook is that all of its graphs are interactive. @media screen and (max-width: 1200px) { Is this a project or just a subjunctive situation? Every 2D matrix-vector multiplication looks like the following: You can think of this multiplication as transforming the vector \(\begin{bmatrix}x \\ y\end{bmatrix}\) into the vector \(\begin{bmatrix}ax + by \\ cx + dy\end{bmatrix}\). How can I graph my own things on this?

At a high level, 3D rendering is just a long series of transformations. But a little rotation is enough for now. A couple weeks ago, I was introduced to Desmos. This graph has lots of different ways of plotting 3D objects. This renderer may not be pretty. You can get the length of a list by using the special \(\length\) function: \(\length([e, \pi, i])\) yields \(3\). In our [x, y, x, y, …] example, the odd-numbered elements are x’s, and the even-numbered elements are y’s. The long version will have to wait for another blog post, but if you’re impatient, just look up “homogeneous coordinates”. A vector field on is a function that assigns to each point a three-dimensional vector . If the resultant is \( \textbf{c} \), then. Source code available at In the usual notation, we can just add more rows, but in Desmos we can do the same by stacking piecewise functions together: Piecewise functions will be a very helpful building block for us because they will help us do different things for the x’s and y’s. Author: Juan Carlos Ponce Campuzano. This video series changed my entire perception of linear algebra and got me hooked on trying to explain topics with visuals and animations. 1 5.

3D Plotter v1.0. This is very similar to our last step - but there’s a twist. Click on the equation to view the variables and constant. I first rotate the model about the X axis, then about the Y axis - you could do the rotations in a different order, but I liked the results I got with that.

Workaround 2: pack everything into one big list. A 3D version of the calculator would be great - we don't have that feature built in yet, but hopefully someday soon! Also @SaeidNourian thanks for that link!

Center: 1 0. This might seem like a surprising step, but it makes the next stage much easier. Marc ten Bosch is developing a video game set in a four-dimensional world. © 2020.

Click on the point plotter button to add a point. A story of four dimensions.

flex-wrap: wrap; But the important thing is that we’ve finally come up with a way to do the math we need. But here’s my whirlwind summary of the parts we’ll be using: We’ll use vectors to represent the points of our model. Perhaps we could just put all the x’s in one list, all the y’s in another, and all the z’s in another still. But Desmos has an important limitation that will make life more difficult: it does not allow lists of lists. It can plot lots of different types of functions in a really intuitive way. You can read Desmos’s out loud like “if x is less than n, the value is sin(x), otherwise, it’s -x”. The simplest way would be to simply go \([x, y, z, x, y, z, \ldots]\). Setting a=2 will show what the graph looks like when y is kept constant at 2, and shows the change in x. Also, it probably seems strange to use four-dimensional vectors to represent three-dimensional data, but the reason we do it is pretty interesting.

Although our points are all three-dimensional, we actually will be using four-dimensional vectors to represent them.

For example, a 3D point with an X position of 1, a Y position of 2, and a Z position of 3 would look like: We’ll use matrices to represent transformations. Now let’s get into the nitty-gritty of how to do this. Another 3blue1brown series, this time on another mathematical construct that is commonly used in 3D graphics applications to make it easier to deal with rotations. The first element of the list will be the first element of \(l\). More precisely, we need to represent vectors and multiplying those vectors by matrices. As cool as the workaround is, it's nice to not need a workaround. We now have to go from world space to camera space, where the camera is at the center. Finally, let’s think about what this list actually looks like.

By Jez Swanson. This means that we need to find the transformations that would move the camera to the right place, but then do those transformations in reverse.


It doesn't show the 3D graph, but does show the x and y slices for a point on a multivariable graph.

What are quaternions, and how do you visualize them? For example, we could make a list of numbers like \([1, 2, 3]\), or a list of 2D points like \(\left[(1, 1), (2, -1), (-1, -1)\right]\). Rotation, transformation, projection - we pretty much just do all our transformations in order and plot the result. Topic: Calculus, Differential Equation. The trick to perspective projection is to imagine stretching and squishing the ends of the frustum so that they’re the same size.

} CCA2 2.1.3: 2-30 Transforming Parabolas (Desmos), CCA2 6.1.2: 3D Point Plotter and Graphing Linear Equations in Three Variables (CPM), CCA2 7.1.3: 7-33 & 7-34 Student eTools (Desmos), CCA2 7.1.3: 7-34 Unit Circle Showing Sine Curve (Desmos), CCA2 7.1.4: 7-52 Student eTool: The Cosine Calculator eTool (Desmos), CCA2 7.1.4: 7-52 Student eTool and Unit Circle Showing Bases and Cosine Curve (Desmos), CCA2 7.1.6: Unit Circle with Reference Triangles (Desmos), CCA2 7.2.1: Transforming Sine and Cosine Functions (Desmos), CCA2 7.2.4: General Form of the Sine Function with Sliders (Desmos), CCA2 8.3.1: Polynomial Division with and without a remainder Videos, CCA2 10.3.1: 10-133 Student eTool (Desmos), CCA2 11.1 & 11.2: Random Number Generator online, CCA2 11.2.1: online & 11-37 eTool (CPM) & Data Representations (CPM), CCA2 11.2.4 7-73 & 7-74 Student eTools (Desmos), CCA2 12.1.4: 12-61 (1/sinx) Student eTool (Desmos), CCA2 12.1.4: 12-65 (1/cosx and 1/tanx) Student eTool (Desmos), CCA2 B.1.4: B-40 & B-41 Student eTool (Desmos).
Once we’re in projection space, we are working in two dimensions!

But it’s a testament to the power of Desmos that it can support this kind of project at all. In order to render an image, we need to figure out how to represent both vectors and matrices in Desmos. Width: 1.

As a counterpoint to 3blue1brown’s quaternion videos, Marc has also written this interactive article promoting the use of rotors instead of quaternions. If you liked this article, you may like some or all of these: If you want to put this math knowledge to use, start here. Now we can flatten the points down along the Z axis to be left with a lovely 2D image: And that’s it! The coordinates seem to be calculated correctly, but the placement in the coordinate system turns out wrong: Well, after redrawing the line the point of intersection seems to be right on spot: @DarbyRathbone That is an amazing tool, thanks so much! We’ll need to try another approach. Maths Geometry Graph plot vector. Has anyone here played Eye Wire before? Sorry its extremely laggy - turns out doing this requires a lot of calculating.

In our case, we use vectors to represent the points of our model and matrices to represent ways of transforming them.

You can see that the model is rotated to a new orientation, and our camera has been moved and rotated to point at it. $$.

For example, look what happens when we use a list on the bottom of our sum: As you can see, Desmos again turns the whole result into a list, but this time turns it into a list of individual sums. I’m omitting the details, but for example, a matrix that rotates vectors by 90° around the y axis would look like: Multiplying a vector by a matrix transforms that vector accordingly. If you’ve never encountered vectors before, the best way to get up to speed is 3blue1brown’s excellent Essence of Linear Algebra series on Youtube.

A couple weeks ago, I was introduced to Desmos.

Similarly, when we’re looking at a y, \(l[n]\) refers to the y, and \(l[n - 1]\), the element before it in the list, is the x. For real-time rendering of 3D equations you can try this:

Maybe in a year’s time I’ll have made an entire game in here. The second element of the list will be the second element of \(l\). $$ max-width: 360px;

This came out while I was working on this article and it made me feel massively inferior. The thing that rules about it is that I can replace your function with what I want and all the stuff you designed still works on it. 2 6. With something like this:, You can even plot the solution to an equation f(x, y, z) = 0. To create a list, you simply put a list of numbers in brackets, like \([1, 2, 3]\). Word of warning: OpenGL is on its way out because of Vulkan (a successor standard) and Apple’s Metal. The reason why is pretty simple - mathematical functions can take many values as input, but produce only one value as output.