# What is an efficient way of drawing surfaces in multivariable calculus?

I've noticed that some surfaces are difficult to draw in multivariable calculus. For instance, I always have trouble with hyperbolic paraboloids.

What is an efficient way to draw the following surfaces:

1)hyperbolic paraboloid

2)hyperboloid (of one and of two sheets)

3)a plane (harder than it seems! for me, at least)

4)elliptic paraboloid

5)ellipsoid

Any other common surfaces you have tricks for would be helpful as well.

• This question was inspired by a similar question for curves in the plane: matheducators.stackexchange.com/questions/7825 Apr 13, 2015 at 16:14
• Thirty years ago we had to draw these by hand with pencil and paper. I recall many a sunny afternoon spent lying in the grass grumbling about needing artistic ability for a math class. Are you looking for ways to make the hand-drawing easier or more accurate, or are you open to software solutions as well? I'm asking for clarification because the link in your comment mentions some software for that similar question. Apr 13, 2015 at 16:33
• @shoover Good question! I am specifically looking for hand drawn-techniques, as many of my classrooms have no computer available. Apr 13, 2015 at 16:35
• I think, and this is not an answer just yet, the key is to add axes at the end. I find this simple rule helps me with most of the things I foolishly attempt to sketch. Great question. Apr 13, 2015 at 17:04
• I start with a piece of scratch paper, and draw some cross sections. In class, I might either use something prepared, or I might communicate the process by starting with a scratch board (and telling the students this is a rough draft), drawing the cross sections I liked, and then transferring the result and experience to the place where I want it on the board. Gerhard "Graphic Mistakes Are Also Allowed" Paseman, 2015.04.13 Apr 13, 2015 at 18:46

(This is not bounty-worthy; just consider these illustrated comments.)

Smith & Minton's textbook Calculus emphasizes drawing curves in each coordinate plane, and a few cross-sections (echoing Gerhard Paseman).

To get an idea of what the graph looks like, first draw its traces in the three coordinate planes.

Although they still rely on software, they talk through the process of gradually coming to understand what particular surfaces should look like.

Here is a (long!) video of Michael Chamberlain sketching surfaces in class. At the below snapshot, he explains how the cross-sections of the hyperboloid are ellipses:

• @MichaelE2: It seems quite natural for people to interpret perspective drawings, as in the snapshot above, where both $x$ and $y$ axes are drawn in a slanted plane, and $z$ vertical. May 3, 2015 at 13:27
• Children learn to interpret pictures of piles of blocks drawn with oblique projection from a young age in math class. I hazard a guess it's natural for them to interpret other pictures drawn that way perfectly easily. The advantage of oblique projection over proper perspective drawing is that parallel lines on the page correspond to parallel lines in space. May 9, 2015 at 5:25

Below are instructions for drawing various quadric surfaces. They're not sketches but drawings designed to look (sort of) pretty and be easy to draw. You can see all of them in one YouTube playlist if you want: https://www.youtube.com/playlist?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo

Hyperbolic paraboloid

https://youtu.be/ZofoRuDJXhM?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo

When you cut a hyperbolic paraboloid with a circular cutter, the outside edge is two cycles of a cos/sin curve. Slices parallel to the x axis and y axis will be parabolas, facing up in one direction and down in the other.

1. Draw a shallow parabolic arc.
2. The left-end is the peak of the sin/cos curve. Draw the down-curve at the back.
3. Draw the down-curve at the front, looping around to reach a peak at the other end of the original parabolic arc.
4. Draw the down-curve at the back, flattening out a little as you reach the middle.
5. At the bottom of the original parabolic arc, draw one side of a down-facing parabolic arc.
6. Fill in the down-facing parabolic sections.
7. Fill in the up-facing parabolic sections.

Hyperboloid of one sheet - emphasis on rotation

https://youtu.be/zQ127ElV6Ns?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo

When you view a hyperboloid of one sheet from the side, you can see a hyperbola. Horizontal cross sections are ellipses and sections through rotating vertical planes are also hyperbolas.

1. Draw a hyperbola opening sideways - remember hyperbolas are almost straight at the edge and turn quite sharply in the middle.
2. Draw an ellipse at the top, and an ellipse arc at the bottom.
3. Fill in horizontal sections - it works well to start in the middle.
4. Fill in rotating vertical sections - a vertical line in the middle and a hyperbolic arc on each side.

Hyperboloid of one sheet - emphasis on vertical sections

https://youtu.be/2XNxBGmtItE?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo

When you view a hyperboloid of one sheet from the side, you can see a hyperbola. Horizontal cross sections are ellipses and sections in vertical planes are hyperbolas in different directions or just two lines.

1. Draw a hyperbola opening sideways - remember hyperbolas are almost straight at the edge and turn quite sharply in the middle.
2. Draw an ellipse at the top, and an ellipse arc at the bottom.
3. Fill in horizontal sections - it works well to start in the middle.
4. Draw two lines crossing on the centre horizontal section.
5. Draw upper and lower branches of a hyperbola inside the two lines.
6. Draw left and right branches of a hyperbola outside the two lines.

Hyperboloid of two sheets - emphasis on rotation

https://youtu.be/XpsgENVJuGA?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo

When you view a hyperboloid of two sheets from the side, you can see a hyperbola. Horizontal sections are ellipses and rotating vertical sections are hyperbolas, with one branch on each nappe of the hyperboloid.

1. Draw a hyperbola opening upward - remember hyperbolas are almost straight at the edge and turn quite sharply in the middle.
2. Draw an ellipse at the top, and an ellipse arc at the bottom.
3. Draw in ellipse horizontal sections on the two nappes.
4. Fill in rotating vertical sections on the bottom nappe - they all cross at a point, and are only really curved at the very top.
5. Fill in rotating vertical sections on the top nappe - you can't see where they cross.

Hyperboloid of two sheets - emphasis on vertical sections

https://youtu.be/kjuJ1pOTiZI?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo

When you view a hyperboloid of two sheets from the side, you can see a hyperbola. Horizontal sections are ellipses and vertical sections are hyperbolas, with one branch on each nappe of the hyperboloid.

1. Draw a hyperbola opening upward - remember hyperbolas are almost straight at the edge and turn quite sharply in the middle.
2. Draw an ellipse at the top, and an ellipse arc at the bottom.
3. Draw in ellipse horizontal sections on the two nappes.
4. Fill in sections on the bottom nappe - resist the urge to make them curved except at the very top.
5. Fill in sections on the top nappe - try to line them up with the sections on the bottom nappe.

Elliptic paraboloid

https://youtu.be/QIlz6A4sqy0?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo

When you view an elliptic paraboloid from the side, you see a parabola. Horizontal sections are ellipses and vertical sections are parabolas.

1. Draw a parabola - make it quite pointy.
2. Draw an ellipse at the top, then fill in the horizontal sections.
3. Draw the vertical sections as not-quite-half-parabolas starting with a very tiny one in the corner.

Ellipsoid

https://youtu.be/0_KWp-2PaP4?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo

When you view an ellipsoid from the side, you see an ellipse. All the sections are ellipses.

1. Draw an ellipse.
2. Starting roughly in the middle, draw the bottom of an ellipse. Fill in a curved arc below this.
3. Draw horizontal sections above the middle line. Include more of the end-curve each time.
4. Draw the vertical sections. Include less and less of the end-curve each time.

Elliptic cone -- emphasis on rotation

When you view a cone from the side, you can see two intersecting lines. The horizontal sections are ellipses, and rotating vertical sections through the vertex always give two lines.

1. Draw two intersecting lines.
2. Draw an ellipse at the top and an elliptical arc at the bottom.
3. Fill in the ellipse horizontal sections. It is easier to start near the top on the top nappe and near the bottom on the bottom nappe, so you can follow the curve of the existing lines.
4. Draw straight lines to indicate rotating vertical sections.

Elliptic cone -- emphasis on vertical sections

https://youtu.be/7wHxjqB-H1M?list=PL3MCc7nq_tLEdscIz_YKTy8ZabMM_yTEo

When you view a cone fromt the side, you can see two intersecting lines. The horizontal sections are ellipses, and vertical sections are two lines if through the vertex or hyperbolas if not.

1. Draw two intersecting lines.
2. Draw an ellipse at the top and an elliptical arc at the bottom.
3. Fill in the ellipse horizontal sections. It is easier to start near the top on the top nappe and near the bottom on the bottom nappe, so you can follow the curve of the existing lines.
4. Draw a line through the vertex off-centre on the top nappe, and another line on the same side but on the bottom nappe.
5. Draw hyperbolic sections on both nappes. Make them mostly straight except where they meet the edge of the cone.
• +7 for the playlist! Nov 13, 2015 at 4:58
• Good point! Updated with a link to the playlist too. Nov 13, 2015 at 5:03

Use level curves and contour plots (click link for video examples of their use). With contour plots, use color to represent third dimension. Here's a familiar example: :