{"id":9898,"date":"2014-11-28T13:15:22","date_gmt":"2014-11-28T05:15:22","guid":{"rendered":"http:\/\/www.intmath.com\/blog\/?p=9898"},"modified":"2014-11-28T14:01:57","modified_gmt":"2014-11-28T06:01:57","slug":"geogebra-now-3d-graphs","status":"publish","type":"post","link":"https:\/\/www.intmath.com\/blog\/mathematics\/geogebra-now-3d-graphs-9898","title":{"rendered":"GeoGebra now with 3D graphs"},"content":{"rendered":"

GeoGebra<\/a> released version 5 a few months back. GeoGebra is a powerful and free graphing tool that anyone learning - or teaching - mathematics would find useful. <\/p>\n

For me, the best feature of the new version is the ability to create 3D graphs.<\/p>\n

When you first open GeoGebra now, you are greeted with this choice of Perspectives:<\/p>\n

\"3D<\/p>\n

Choosing "3D Graphics", you get several new panels, which allow you to create 3D objects like a line perpendicular to a plane, a plane intersecting a cone, a plane through 3 points, a sphere, and so on:<\/p>\n

\"3D<\/p>\n

You also get a set of empty 3-D coordinate axes, like this:<\/p>\n

\"3D <\/p>\n

Of course, you can still create 2D graphs as before, and the interface is largely unchanged for that aspect.<\/p>\n

Some 3D graph examples<\/h2>\n

Let's draw a few 3D surfaces using GeoGebra.<\/p>\n

Here's a water droplet-like shape, whose equation is: <\/p>\n

z<\/em>(x<\/em>, y<\/em>) = 1 + 3 cos((x<\/em>2<\/sup> + y<\/em>2<\/sup>) 2) e<\/em>(-(x<\/em>2<\/sup> + y<\/em>2<\/sup>;))<\/sup><\/p>\n

\"water <\/p>\n

Next, here are some 3D graphs that were suggested by some comments on the article How to draw y^2 = x - 2?<\/a> <\/p>\n

One reader asked how to draw some graphs involving interesting asymptotes. I used some different software for those ones (which has a problem where asymptotes appear as vertical "walls" - but shouldn't be there at all.<\/p>\n

Those graphs are similar to the following ones, which also involve an asymptote: <\/p>\n

z<\/em> = y <\/em>\/ x<\/em>2<\/sup><\/span><\/p>\n

\"3D<\/p>\n

Let's consider this curve for a bit (using GeoGebra to help, of course), and investigate it from different angles. <\/p>\n

If we fix y<\/em> = 1<\/span> then the curve z<\/em> = 1 \/ x<\/em>2<\/sup><\/span> (it's in 2 dimensions) looks like this: <\/p>\n

\"3D<\/p>\n

We can see this shape in our 3D graph above.<\/p>\n

Let's now add the negative of the above graph, that is z<\/em> = −y <\/em>\/ x<\/em>2<\/sup><\/span><\/p>\n

Here's what it looks like, along with our original surface:<\/p>\n

\"3D <\/p>\n

Next, we'll look at z<\/em> = −x <\/em>\/ y<\/em>2<\/sup><\/span> by itself:<\/p>\n

\"3D<\/p>\n

Now, we add its negative: z<\/em> = −y <\/em>\/ x<\/em>2<\/sup><\/span> (the one in green): <\/p>\n

\"3D<\/p>\n

Now for all 4 surfaces together:<\/p>\n

\"4<\/p>\n

Here they are from a different point of view:<\/p>\n

\"another <\/p>\n

Conclusion<\/h2>\n

GeoGebra is a versatile and powerful tool. With this new version, they have addressed some of the shortfalls of earlier versions, since it now handles 3D graphs, and there are non-Java output possibilities so you can view the applets, and interact with them, on your tablet devices.<\/p>\n

Trying to imagine surfaces in 3D really pushes your brain envelope - but this tool makes things a lot easier!<\/p>\n

Sweet! <\/p>\n

See the 11 Comments<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"

\"3D<\/a>
GeoGebra's latest version offers 3D graphs and output that can be viewed on tablet devices. Here's a quick overview.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mo_disable_npp":""},"categories":[4],"tags":[109,127],"_links":{"self":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/9898"}],"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\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/comments?post=9898"}],"version-history":[{"count":0,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/9898\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=9898"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=9898"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=9898"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}