{"id":10759,"date":"2016-04-13T17:52:23","date_gmt":"2016-04-13T09:52:23","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=10759"},"modified":"2016-04-16T10:28:59","modified_gmt":"2016-04-16T02:28:59","slug":"ew-3d-interactive-graph","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/ew-3d-interactive-graph-10759","title":{"rendered":"New 3D interactive graph"},"content":{"rendered":"

Most students struggle when trying to understand 3D cartesian coordinates, especially when diagrams in text books are static, and in 2 dimensions only (of course). <\/p>\n

I recently published a new 3D cartesian coordinates interactive graph:<\/p>\n

3D Space Interactive Applet<\/a><\/p>\n

In the introduction to 3D graphs, I gave the example of the point P (2, 3, 5)<\/span> . The 3D interactive graph allows you to explore what a point in 3 dimensions means and helps to orientate those unfamiliar with the xyz<\/em>-coordinate system.<\/p>\n

You can vary the x<\/em>-, y<\/em>- and z<\/em>-coordinate of the given point and rotate the axes, as well as zoom in and out. <\/p>\n

This is quite a simple interactive and I hope to introduce more involved ones in the coming months.<\/p>\n

I developed the applet using Three.js<\/a>, a 3D javascript library. As such, the applet is cross-platform, cross-browser and mobile friendly.<\/p>\n

xyz<\/em> axes orientation?<\/h2>\n

When I first wrote the section on 3D Space<\/a> in the Vectors<\/a> chapter, I wanted to ensure I chose a "good" orientation for the 3 axes. By "good", I mean that I was using a proper convention and that it was commonly used.<\/p>\n

For interest, I looked at what Google Images returned for a search on xyz<\/em> axes. I'm using the results of that search below. <\/p>\n

When I first learned 3D geometry in secondary school, we started with the familiar xy<\/em>-axes and drew the new z<\/em>-axis coming "out of the page" towards us, like this (let's call it Orientation #1<\/strong>): <\/p>\n

\"xyz<\/p>\n

At times we would see the axes rotated around the y<\/em>-axis, like this one (let's call it Orientation #2<\/strong>):<\/p>\n

\"xyz<\/p>\n

But that was OK - the y<\/em>-axis was still at the top, so we didn't get lost. <\/p>\n

When I met the idea again at university, the following convention was used. The z<\/em>-axis was vertical, with the x<\/em>-axis going to the left (and down-ish), and the y<\/em>-axis going to the right (also down-ish), like this:<\/p>\n

\"xyz<\/p>\n

This change of orientation was quite disconcerting at first. I kept drawing points, planes and graphs the wrong way round.<\/p>\n

Now I'm more used to this orientation (let's call it Orientation #3<\/strong>) , and it seems that it's more common to have the z<\/em>-axis going up.<\/p>\n

You'll also see it with the x<\/em>- and y<\/em>-axes rotate 90°, like this (let's call it Orientation #4)<\/strong><\/strong>: <\/p>\n

\"xyz <\/p>\n

Right-hand Rule<\/h3>\n

Each of the above orientations obeys the right hand rule<\/strong>, whereby if our right-hand index finger points in the direction of the x<\/em>-axis, and our second finger points in the direction of the y<\/em>-axis, then our thumb will point to the z<\/em>-index.<\/p>\n

\"right<\/p>\n

(Yes, you have to contort your hand around somewhat to ensure each of the above orientations work, but they do.) <\/p>\n

The following one (found in Google's index of such axes) fails the right-hand rule:<\/p>\n

\"xyz <\/p>\n

(To be fair, it is possible the z<\/em>-axis is coming out of the page towards us, but that's not the impression I get.) <\/p>\n

Is a vertical z<\/em>-axis more common?<\/h3>\n

The calculus text books by Kreyszig<\/strong>, Washington, Mizrahi\/Sullivan<\/strong> and Thomas\/Finney<\/strong> all use Orientation #3, which is why I used that orientation as well.<\/p>\n

Wolfram Alpha<\/strong>'s default 3D view is the same as Orientation #4 (where the positive x<\/em>-axis is to the right and down, and the y<\/em>-axis is to the right and up.) <\/p>\n

Here's the curve z<\/em> = x<\/em>2<\/sup> + sin(y<\/em>)<\/span> by Wolfram|Alpha: <\/p>\n

\"Graph<\/p>\n

Geogebra<\/strong> has the same default orientation. Here's the same graph using Geogebra, which doesn't appear to have an option for naming the axes (it does for 2D axes). <\/p>\n

\"Graph<\/p>\n

Up next is Scientific Notebook<\/strong>, which uses x<\/em> to the right and down, and y-axis to the left and down, with z<\/em> up.<\/p>\n

This means it fails<\/strong> the right hand rule! At least the axes are labelled. <\/p>\n

\"Graph<\/p>\n

Matlab<\/strong>'s plot on first glance appears to be the same as Wolfram|Alpha's, but in fact, the positive y<\/em>-axis is going off to the left and up.<\/p>\n

It obeys the right hand rule, but this is not a common orientation.<\/p>\n

\"xyz<\/p>\n

Why does it matter?<\/h2>\n

As I mentioned earlier, a lot of novice learners have difficulty imagining planes and other surfaces in 3D. Presenting graphs in a myriad of orientations doesn't help.<\/p>\n

Suggestions for teachers <\/h2>\n
    \n
  1. The first examples of 3D graphs that students see should use a consistent orientation, as far as possible (of course, we need to turn things around a bit to see key features, but keep such turns minimal)<\/li>\n
  2. Slowly introduce other orientations, and point out clearly the features of the new orientation and why you are using it<\/li>\n
  3. Always label the axes<\/strong> and insist students do so as well. If you are using one orientation, and they understand it to be another, confusion is sure to follow<\/li>\n
  4. Eventually give examples of all the possible 3D axes orientations, so students can figure out the differences. <\/li>\n<\/ol>\n

    The link to the interactive graph again:<\/p>\n

    3D Space Interactive Applet<\/a><\/p>\n

    Be the first to comment<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"

    \"3D<\/a>
    \nI recently added a 3D interactive graph which you can use to explore the x<\/em>-, y<\/em>- and z<\/em>-axis system.<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mo_disable_npp":""},"categories":[4],"tags":[134,127],"_links":{"self":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/10759"}],"collection":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/comments?post=10759"}],"version-history":[{"count":0,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/10759\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=10759"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=10759"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=10759"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}