Projective geometry

#1644

From: <arthur.lawrence@...>
Date: Sat Oct 3, 2009 1:57 am
Subject: FORGOT MONKEY FIST

artplot101

FORGOT TO ADD ATTACHMENT (DUH)

ARTPLOT

1 of 1 File(s)

MONKEY FIST01.gcf

#1645

From: <arthur.lawrence@...>
Date: Sat Oct 3, 2009 1:53 am
Subject: MONKEY FIST

artplot101

FOOLING AROUND WITH TORI AND CAME UP WITH THIS!

ARTPLOT

#1646

From: "helper" <website.reader3@...>
Date: Mon Nov 9, 2009 9:59 pm
Subject: trying to create a gradually drawn 3d curve

website.reader

I have a simple problem which still escapes me.  I am trying to draw a 3d space
curve, using n, instead of t, which slowly grows, like a trajectory path, as n
increases.

When I use a 3 vector format for the x,y,z positions, graphing calculator draws
in a whole bunch of points simultaneously, as if it is attempting to use all
possible values of n at once.

Any ideas?

I am trying to do a gradually drawn helix.

Reply | Messages in this Topic (3)

#1647

From: David Craig <craigda@...>
Date: Mon Nov 9, 2009 10:48 pm
Subject: Re: trying to create a gradually drawn 3d curve

dcraigx

On Nov 9, 2009, at 4:59 PM, helper wrote:

> I have a simple problem which still escapes me. I am trying to draw
> a 3d space curve, using n, instead of t, which slowly grows, like a
> trajectory path, as n increases.

Use n*t as the parameter.   GC will plot the t from 0 to 1 (or
whatever range you set for t); as n grows the plotted parameter range
goes from 0 to n*t.

David Craig

<http://web.lemoyne.edu/~craigda/>

Reply | Messages in this Topic (3)

#1648

From: Chris Young <ChristopherYoung56@...>
Date: Tue Nov 10, 2009 6:36 am
Subject: Re: trying to create a gradually drawn 3d curve

c1572young

Here's a simple example where you can have both the regular curve via F(t) and the progressively drawn curve via F(nt), where n runs from 0 to 1.

On Nov 9, 2009, at 5:48 PM, David Craig wrote:

On Nov 9, 2009, at 4:59 PM, helper wrote:

> I have a simple problem which still escapes me. I am trying to draw
> a 3d space curve, using n, instead of t, which slowly grows, like a
> trajectory path, as n increases.

Use n*t as the parameter. GC will plot the t from 0 to 1 (or
whatever range you set for t); as n grows the plotted parameter range
goes from 0 to n*t.

David Craig

2 of 2 Photo(s)

Progressively drawn curves via n slider.gcf

1 of 1 File(s)

Reply | Messages in this Topic (3)

#1649

From: Chris Young <ChristopherYoung56@...>
Date: Sat Nov 21, 2009 6:46 am
Subject: Parametric curves tutorial. Part 1.

c1572young

The basic idea behind parametric curves is simply to use the reserved variable "t" along with vectors or complex numbers.

Below, t multiplying the [2 1] vector creates a line segment of length 1.

To draw a line segment between two points, use (1 - t) times the start point and t times the end point. The green line segment is the result.

Here's the part of the Help on 2D curves, from "Browse" and then "Graphing Equations", "Two dimensions".

The 3D method is similar.

If you want to extend line segments, you have to either change the t parameter range in the lower right corner or else multiply some factor times t in the equation. I recommend avoiding changing the t range using the range setting at bottom right, because then all your graphs will be affected in that file. I wish we had a way to set the t range for each equation, the way Grapher, Mathematica, etc. do.

To get a curve, just enter in any functions for the vector coordinates.

I think of this as the t parameter being a kind of "tracing pen". Whichever changing quantity you put it next to is the one that will get traced. For the circle and the ellipse above, we wanted to trace the angular motion , so we put the t next to the angle coefficient 2�.

We could just use t without multiplying it be 2�, but then we'll only get a shorter arc, not the whole circle or ellipse (of length a in the case of the circle, I think).

As I said above, you can change the range by putting in different limits in the small "t:0�1" range setting in the bottom right of the equation pane. Just be aware that this changes the t range for every equation in the file. You might not always want to do that.

8 of 8 Photo(s)

#1650

From: Chris Young <ChristopherYoung56@...>
Date: Sat Nov 21, 2009 6:47 am
Subject: Parametric curves, 2. Using "t" parameter to draw different parts of figure. Functions.

c1572young

Here's part 2, dealing with applying the drawing parameter t to different parts of an expression to draw different elements of a figure.

For the circle, if we put the t right next to the radial factor (a in the case of the circle, a and b in the case of the ellipse), then we draw radial line segments outward from the center, rather than the circular arcs we add when the t was the part of the angle argument for cos( ) and sin( ).

Below, I've specified that the rays go out every 45 degrees by using the parameter list

3 of 3 Photo(s)

F1a. Using functions for circle and ellipse.gcf

5 of 5 File(s)

pastedGraphic.tiff

D1b. Circle with _radial_ line segments drawn.gcf

#1651

From: Chris Young <ChristopherYoung56@...>
Date: Sat Nov 21, 2009 6:47 am
Subject: Parametric curves, 3. Connecting corresponding parts of figures.

c1572young

Below, we connect the corresponding dots on the circumscribing circle to those on the ellipse:

By the way, GC has a really irritating bug with function names: If you type a variable right in front of a function, it will get "grabbed" by the function sometime:

You have to put the function in parentheses to avoid this:

Above, we can see that the x coordinate of each point on the ellipse is a * cos(2�k), where k is the list of fractions of the circumference.

How do we get the other coordinate? Well, a reasonable guess, by symmetry principals, is to use the inscribed circle for the ellipse, since we had such good luck with the circumscribed circle.

Below, we've finally solved the mystery of where the angle (we're using t for it) in the expression for the ellipse comes from.

It's the angle along both the inscribed and the circumscribed circles. We drop vertical lines from the circle of radius a (in red), and horizontal lines from the circle of radius b (in blue). These correspond respectively to

and

, the coordinates of the ellipse in terms of the central angle.

Note that even when the "major radius" a is shorter than the "minor radius" b, we have the same situation: take the cosines (multiplied by a) by dropping vertical projection lines from the red circle, and take the sines (multiplied by b) by dropping horizontal projection lines from the blue circle.

http://www.xahlee.org/SpecialPlaneCurves_dir/ggb/ellipse_pointwise_const2.html has an interactive demo, and there is a related property illustrated at http://www.xahlee.org/SpecialPlaneCurves_dir/Ellipse_dir/ellipse.html.

However, I think it's more helpful to follow the above method I used, and think of an ellipse as a circle scaled differently in the two different directions of the coordinate axes.

7 of 7 Photo(s)

#1652

From: Chris Young <ChristopherYoung56@...>
Date: Sat Nov 21, 2009 7:21 am
Subject: Parametric curves, 3. Ellipse as scaled circle. Animation.

c1572young

To finish off the discussion of how we can investigate parametric curves interactively with GC, we can check that an ellipse is simply a scaled circle by applying the matrix for scaling along x and y,

I've used the k parameter-list to plot two new purple ellipses, one just inside and one just outside our original ellipse.

Below, we still see the same overall pattern of projections when the major radius a is greater than the minor radius b. We still drop vertical lines from the red circle and horizontal ones from the blue circle. The intersection points are the points on the ellipse for each given angle.

It would probably be clearer to use black lines for the angle rays, to emphasize that we're using the same angles for corresponding points on the red and blue circles.

Below, I've also added heavier lines around the angle selected by the animation slider, i.e., 2�n, as well as a darkened sector corresponding to the size of the angle.

4 of 4 Photo(s)

H4b. Animation of angle.gcf

4 of 4 File(s)

pastedGraphic.tiff

H2a. Major radius longer than minor..gcf

H1b. Scaling the circle via a matrix.gcf

#1653

From: "n914tj" <N914TJ@...>
Date: Fri Nov 27, 2009 8:47 pm
Subject: I need four more labels!

n914tj

I need 45 labels to individually control 45 chords in a polyhedron.  There are
26 letters in the alphabet.  All 26 of the upper case letters are available, but
only 15 of the lower case letters can be used.  The letters e, i, n, r, t, u, v,
w, x, y and z are all reserved.  This makes a total of 41 available labels. 
Anybody know where I can find four more?  I'm using GC3.2.

#1654

From: Chris Young <ChristopherYoung56@...>
Date: Sat Nov 28, 2009 7:21 am
Subject: Re: I need four more labels! Piecewise defined functions.

c1572young

On Nov 27, 2009, at 3:47 PM, n914tj wrote:

I need 45 labels to individually control 45 chords in a polyhedron. There are 26 letters in the alphabet. All 26 of the upper case letters are available, but only 15 of the lower case letters can be used. The letters e, i, n, r, t, u, v, w, x, y and z are all reserved. This makes a total of 41 available labels. Anybody know where I can find four more? I'm using GC3.2.

If you can upgrade to 3.5 you can use subscripts on the letters to get more variables. I don't think that can be done with GC 3.2. At least I don't see subscripts listed in the listing of features for GC 3.2 that I've saved.

I think GC 3.2 lets you use piecewise defined functions. You could simply pick a numbering scheme for your polyhedron's chords and then assign lengths within the function definition:

I couldn't assign the points via a piecewise defined function, but I could assign them one point at a time:

Another workaround might be to use "dummy function definitions" with the same name. E.g., you could define a function

a( )=4

where 4 (or whatever) is the length of the chord. I didn't even have to have an argument for the function; GC 3.5 automatically saved the value I assigned.

Graphing Calculator 3.2
for MacOS and Windows

If you are not already using either Graphing Calculator 3 or NuCalc 2.0 go here to order Version 3.2.

If you have already purchased Graphing Calculator 3 for Macintosh or NuCalc 2 for Windows go here to upgrade to version 3.2.

To be notified of new versions when they are available, sign up for one of these mailing lists.

What's new in Version 3.2

* Common features and file format on Mac and Windows
* Free Viewer application available on Mac and Windows
* Vectors
* Coordinate Transformations

------------------------------------------------------------------------
What was new in Version 3.1

* Complex numbers
* Complex number arithmetic
* Complex function parametric curves in 2D
* Complex function parametric curves in 3D
* Complex function surfaces in 3D
* Complex function surfaces in 3D with color coding
* Complex function surfaces in 4D
* Interactive complex parameters
* Color maps in 2D
* Point plots in 2D
* Vertex lists in 4D
* Parametric curves in 4D
* Parametric surfaces in 4D
* Examples menu and sample documents
* New functions
* min(a,b)
* max(a,b)
* mod(a,b)
* clamp(x,a,b)
* Text expressions in coordinate ranges
* Direction fields

What was new in Version 3.0

* Text comments can be added to documents
* Printing
* Open & Save Graphing Calculator documents
* Multiple document windows
* Save for Web saves HTML files and PNG images
* Save as RTF for export to word processors
* 3.0 acts as a helper application for GC documents posted to the web
* Documents saved by 3.0 can be opened read-only by the free version 1.2.

What was new in Version 2.7

* Ability to change the 3D background color
* Ability to define functions and variables symbolically
* Ability to copy and paste formulas as plain text
* Equations now line-break and scroll
* 1st order ordinary differential equations in 2 dimensions
* Correct handling of fractional powers
* New functions
* Bessel functions
* floor, ceiling
* Piecewise defined functions
* Div, grad, and curl, vector dot, and cross products
* Symbolic summation notation
* Symbolic substitutions
* Integrals

3 of 3 Photo(s)

B3c. Coudn't assign points via function.gcf

2 of 2 File(s)

A1a. "Dummy functions" as extra variables.gcf

#1655

From: dan@...
Date: Fri Dec 11, 2009 12:33 pm
Subject: off-topic, maybe, but beautiful

dan@...

http://www.wired.com/wiredscience/2009/12/mandelbulb-gallery/

The quest by a group of math geeks to create a three-dimensional
analogue for the mesmerizing Mandelbrot fractal has ended in
success.

They call it the Mandelbulb. The 3-D renderings were generated
by applying an iterative algorithm to a sphere. The same
calculation is applied over and over to the sphere's points in
three dimensions. In spirit, that's similar to how the original
2-D Mandelbrot set generates its infinite and self-repeating
complexity.

If you were ever mesmerized by the Mandelbrot screen saver, the
following images are worth a look. Each photo is a zoom on one
of these Mandelbulbs.

#1656

From: "gjmcclure" <greg_mcclure@...>
Date: Mon Jan 18, 2010 4:21 pm
Subject: Graphs of electron orbitals...

gjmcclure

Recently I found a site that contains the equations to create the graphs for the
electron orbitals (s, p, d, f...) but when I use Graphing Calculator to try to
recreate the orbital graphs shown, I don't get the same results.  The site is...

http://www.uky.edu/~holler/html/orbitals_2.html

The graphs are shown by chosing the orbital in question from the left column,
the equations are shown by clicking on the equations reference on top.

I am making an assumption, which may be the problem... the graph would be of the
form r = f(theta, phi), so I pick the phi from column 1, and theta from column
2.  If I only use column 1 (the phi column) then I get shapes that look like the
0 value orbital for each state (I take the absolute value to get the other half
for some of them).  But I can't get the theta sections to work, they give me
very wierd surfaces that don't look at all like an orbital.

Okay, what am I doing wrong here?

#1657

From: Craig David <craigda@...>
Date: Mon Jan 18, 2010 11:51 pm
Subject: Re: Graphs of electron orbitals...

dcraigx

I'm going to take the easy way out here because I've got about 49 things that
need doing before classes begin.   Have a gander at the "Mathematical Physics"
section of

<http://web.lemoyne.edu/~craigda/Physics/Sims/Files/filelibrary.html>.

I've posted a gc file that plots spherical harmonics (which is what those images
of orbitals really are.)

For a REALLY cool application check out

<http://daugerresearch.com/orbitals/index.shtml>

David




On Jan 18, 2010, at 11:21 AM, gjmcclure wrote:

> Recently I found a site that contains the equations to create the graphs for
the electron orbitals (s, p, d, f...) but when I use Graphing Calculator to try
to recreate the orbital graphs shown, I don't get the same results. The site
is...
>
> http://www.uky.edu/~holler/html/orbitals_2.html
>
> The graphs are shown by chosing the orbital in question from the left column,
the equations are shown by clicking on the equations reference on top.
>
> I am making an assumption, which may be the problem... the graph would be of
the form r = f(theta, phi), so I pick the phi from column 1, and theta from
column 2. If I only use column 1 (the phi column) then I get shapes that look
like the 0 value orbital for each state (I take the absolute value to get the
other half for some of them). But I can't get the theta sections to work, they
give me very wierd surfaces that don't look at all like an orbital.
>
> Okay, what am I doing wrong here?
>
>

David Craig


<http://web.lemoyne.edu/~craigda/>

#1658

From: Christopher Young <cy56@...>
Date: Tue Jan 26, 2010 3:26 pm
Subject: New pages on ruled surfaces on website.

c56young

http://home.comcast.net/~cy56/ws1/Math/mathtoc.html

A few more pages added to my GC website. Still entranced by the problem of ruled surfaces. Amazed that any frame that is strung with the same number of strings on each opposite side will always produce a doubly-ruled surface. Most of these files are GC4 files, and can't be run with GC 3.5. However, I hope that they'll at least get some interest going in how much can be done with GC4.

Clicking on most of the images of files will download a copy.

Chris Young

Christopher O. Young

ChristopherYoung56@...

#1659

From: Christopher Young <cy56@...>
Date: Sun Jan 31, 2010 3:20 pm
Subject: Quicktime movie on website of hyperbolas via "sweep lines"

c56young

Still working out how to relate all these different approaches to conics.
Ultimately, will be in terms of projective geometry and Dandelin spheres, I
think. Probably no other clear way to unite them all.
Will have to shuffle things around a lot.

The good news is that small-size movies are working OK.

http://home.comcast.net/~cy56/ws1/Geometry/Analytic%20Geometry/Curves/index.html

2 of 2 Photo(s)

PastedGraphic-1.png

PastedGraphic-2.png

#1660

From: Christopher Young <cy56@...>
Date: Sun Jan 31, 2010 3:35 pm
Subject: Fixed error in formula for conics via focal parameter

c56young

http://home.comcast.net/~cy56/ws1/Geometry/Analytic%20Geometry/Curves/index.html

Had some non-matching subscripts for the focal parameters k relative to the eccentricities, so the major and minor axes of the ellipse and the hyperbola weren't coming out right.

Fixed below:

#1661

From: Christopher Young <cy56@...>
Date: Sun Jan 31, 2010 3:56 pm
Subject: Re: Projective geometry

c56young

On Jan 31, 2010, at 10:40 AM, Steven Rubenstein wrote:

Chris,

Can you explain projective geometry to me in a nutshell?

Steve

Christopher O. Young

ChristopherYoung56@...

Everything that doesn't change when you project a figure, just as you'd project a slide on a screen.

Except that the projector can go both ways; a double cone.

Instead of angles and distances and so on staying the same, we have perspective properties such as the "cross-ratio" staying the same.

I've got a great diagram from Kleins "Elem. Math: Geometry" that I'm going to upload later. It relates the cross-ratio to the Inscribed Angle Theorem.

There are fewer invariants, but they're more fundamental. Most importantly, the exceptional cases due to points and lines at infinity are brought into the same category as the others.

At least, that's my best idea right now. Felix Klein is the one who gives me the most insights, but I've been rereading his books for years.

Projective geometry
From Wikipedia, the free encyclopedia
In mathematics, projective geometry is the study of geometric properties which are invariant underprojective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, in a given dimension, and that geometric transformations are permitted that move the extra points (called "points at infinity") to traditional points, and vice versa. The properties that matter in projective geometry are those that are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations; the first issue for geometers is what kind of geometric language would be adequate to the novel situation. It is not possible to talk about angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant under projective transformations, as is seen clearly in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point at infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basic (railway tracks meeting at the horizon in a perspective drawing). See projective plane for the basics of projective geometry in two dimensions.
While the ideas were available earlier, projective geometry was mainly a development of the nineteenth century. A huge body of research made it the most representative field of geometry of that time. This was the theory of complex projective space, since the coordinates used (homogeneous coordinates) were complex numbers. Several major strands of more abstract mathematics (includinginvariant theory, the Italian school of algebraic geometry, and Felix Klein's Erlangen programmeleading to the study of the classical groups) built on projective geometry. It was also a subject with a large number of practitioners for its own sake, under the banner of synthetic geometry. Another field that emerged from axiomatic studies of projective geometry is finite geometry.
The field of projective geometry is itself now divided into many research subfields, two examples of which are projective algebraic geometry (the study of projective varieties) and projective differential geometry (the study of differential invariants of the projective transformations).

#1662

From: Steven Rubenstein <SJRubenstein@...>
Date: Sun Jan 31, 2010 3:40 pm
Subject: Re: Quicktime movie on website of hyperbolas via "sweep lines" [2 Attachments]

akaroy

Chris,

Can you explain projective geometry to me in a nutshell?

Steve


Christopher Young wrote:
> Still working out how to relate all these different approaches to conics.
> Ultimately, will be in terms of projective geometry and Dandelin
> spheres, I think. Probably no other clear way to unite them all.
> Will have to shuffle things around a lot.
>
> The good news is that small-size movies are working OK.
>
>
http://home.comcast.net/~cy56/ws1/Geometry/Analytic%20Geometry/Curves/index.html
>
>

#1663

From: Christopher Young <cy56@...>
Date: Sun Jan 31, 2010 11:17 pm
Subject: Re: Applications of projective geometry

c56young

Physical Applications of Geometric Algebra

On Jan 31, 2010, at 11:50 AM, Steven wrote:

What's the point?

On Jan 31, 2010, at 10:40 AM, Steven Rubenstein wrote:

> > Chris,
> >
> > Can you explain projective geometry to me in a nutshell?
> >

> > Steve

At this point, I can't give you a direct answer. But I think that's what physicists probably said about complex arithmetic and analysis, and now it's used throughout physics. You just mentioned that Sachs's theory was based on using quaternions.

Understanding the geometry of orbitals in a more unified way has got to be a help throughout physics.

Computer graphics would be much slower and more difficult without it.

Here's something I came across doing a Google search for "physical applications of projective geometry":

#1664

From: Christopher Young <cy56@...>
Date: Sun Jan 31, 2010 11:20 pm
Subject: Applications of projective geometry to relativity theory

c56young

[PDF]
Projective geometry and special relativity
File Format: PDF/Adobe Acrobat - Quick View
by D Delphenich - 2005 - Cited by 3 - Related articles
[22] O. Conradt, The principle of duality in Clifford algebra and projective geometry, in. Clifford Algebras and their Applications in Mathematical Physics, ...
arxiv.org/pdf/gr-qc/0512125
Clifford algebra, geometric algebra, and applications
Projective and conformal geometry. Various applications in physics (classical mechanics, electromagnetism, special relativity / Minkowski space, ...
www.math.kth.se/~dogge/clifford/ - Cached - Similar

From the first paper:

#1665

From: Christopher Young <cy56@...>
Date: Mon Feb 8, 2010 2:03 pm
Subject: New web pages on implicit differentiation; one-sided surfaces

c56young

The table of contents is at http://home.comcast.net/~cy56/ws1/Math/mathtoc.html
and some of the new pages are at

http://home.comcast.net/~cy56/ws1/Math/Topology/mobius1.html
http://home.comcast.net/~cy56/ws1/Math/Topology/mobiusstripswith.html
http://home.comcast.net/~cy56/ws1/Math/Calculus/implicitdifferen.html

#1666

From: Christopher Young <cy56@...>
Date: Mon Feb 8, 2010 3:22 pm
Subject: New web page on M�bius strip "space station"

c56young

http://home.comcast.net/~cy56/ws1/Math/Topology/index.html

The idea is to make the concept of an intrinsic test of non-orientability clear
and vivid.
I'm a little worried about how I'm constructing the "space station", via
sweeping a fixed-size rectangular frame around.

It doesn't work too well when the surface self-intersects, since the walls of
the "space station" get pinched to zero width along certain lines. Back to the
drawing board as far as space station design goes. The big question is: Is it
possible to construct a self-intersecting one-sided "surface" with thickness?

These are files made with Graphing Calculator 4, still in development. However,
they're intended to show just how much GC4 is capable of, with minimal effort
compared to something like Mathematica when it comes to vivid, interactive
graphics with straightforward, mathematical specifications.

1 of 1 Photo(s)

PastedGraphic-5.png

#1667

From: Christopher Young <cy56@...>
Date: Mon Feb 8, 2010 3:35 pm
Subject: New web page on M�bius strip "space station", link corrected

c56young

Sorry, that link is now:

http://home.comcast.net/~cy56/ws1/Math/Topology/mobiusSpaceStn.html

The idea is to make the concept of an intrinsic test of non-orientability clear
and vivid.
I'm a little worried about how I'm constructing the "space station", via
sweeping a fixed-size rectangular frame around.

It doesn't work too well when the surface self-intersects, since the walls of
the "space station" get pinched to zero width along certain lines. Back to the
drawing board as far as space station design goes. The big question is: Is it
possible to construct a self-intersecting one-sided "surface" with thickness?

These are files made with Graphing Calculator 4, still in development. However,
they're intended to show just how much GC4 is capable of, with minimal effort
compared to something like Mathematica when it comes to vivid, interactive
graphics with straightforward, mathematical specifications.

#1668

From: Christopher Young <cy56@...>
Date: Sun Mar 7, 2010 6:19 am
Subject: Quick ways to put grids on surfaces

c56young

A quick easy way to get a grid is simply to select the "checkerboard" pattern:

You can increase the fineness of the grid via the resolution slider in the lower
left corner of the window:

If you want to get individual grid lines, you can just substitute in the "t"
parameter for one of the "u" or "v" parameters, and use a variable and a
parameter list for the other parameter. Below, I multiplied the ellipsoid
function by a number slightly greater than 1 to get the lines spaced out from
the ellipsoid so that they get graphed more neatly, although that's not strictly
necessary.

3 of 3 Photo(s)

1 of 1 File(s)

#1669

From: Christopher Young <cy56@...>
Date: Wed Mar 10, 2010 6:43 am
Subject: Tips on drawing with the "t" parameter

c56young

Two-dimensional parametric plots;1.gcf

Here are some tips on how to use the "t" parameter to draw lines in 2D and 3D.

First, check out the Help section relating to Two-dimensional graphs:

Here's a file with the examples above plotted:

4 of 4 Photo(s)

2 of 2 File(s)

#1670

From: Christopher Young <cy56@...>
Date: Sat Mar 20, 2010 3:31 am
Subject: xy grid superimposed on a surface

c56young

B1a. Checkerboard pattern.gcf

What I want to be able to do is to see the x y grid superimposed onto the z axis,like the examples in the calculus textbook.

I think you must mean superimposed on the surface, right?

You can automatically put check marks which follow the xy grid just be selecting the checkered pattern:

One possible limitation of the pattern is that as the checks are made bigger (via the resolution slider), the surface becomes more jagged:

So, to overcome that, we can draw gridlines ourselves. In GC 3.5 we can use a parameter list

to graph each set of grid lines (parallel to the x axis and parallel to the y axis).

But, as I remember, for the latest Windows version we don't have parameter lists, so we have to plot each grid line separately:

5 of 5 Photo(s)

5 of 5 File(s)

C1b. Parameter lists.gcf

B2a. Bigger checks means less smoothness.gcf

A2a. Not drawing two curves above and below, just one on the surface.gcf

PastedGraphic-7.tiff

#1671

From: Christopher Young <cy56@...>
Date: Sat Mar 20, 2010 3:44 am
Subject: Gridlines on surfaces, using defined function for the surface

c56young

The whole process of putting gridlines on a surface becomes a lot simpler if we
put our surface into a function. Then we can define separate parameters using
"X" and "Y" with different subscripts. In every case, we're just plotting a
vector function of the form vector(x,y,z)=vector(X,Y,function(f,X,Y)). (You can
copy that last expression directly if the following graphic doesn't show up.)

#1672

From: Christopher Young <cy56@...>
Date: Sat Mar 20, 2010 3:57 am
Subject: Beware the grabby trigs!

c56young

We have to watch out that GC doesn't grab expressions after the trig functions
and exponential function.
I personally would rather it wasn't quite so grabby. When typing a factor after
a trig function, be sure to enclose the trig function in parentheses so it
doesn't grab what follows as part of the argument. It's not enough just to type
an asterisk for multiplication (the way I wish GC behaved.)