1. EXPLICIT 3D EQUATIONS
z = f(x, y)

Tutorial recording || Autograph file
Here a single point is placed on the function
z = asinxcosy, which is in turn intersected with z = 0.

Select the point and use the right-click option “Unit Normal Vector”, and “Tangent Plane” (small).

The point can be dragged around the surface (or moved with the arrow keys, varying ‘x’ and ‘y’ independently), to investigate the local max and min in 3D.

2. IMPLICIT 3D EQUATIONS - f(x, y, z) = 0
Tutorial recording || Autograph file

Autograph can plot any implicit equation,
e.g. Schwarz's P surface: cosx + cosy + cosz = 0.

Other possibilities are:
Sphere: x² + y² + z² = r²
Cylinder: x² + y² = r²
Box: |x| + |y| = r

3. . THE CONIC SECTIONS
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Cone: x² + y² = z²
[can also be input as r = z, using ]

Intersections with a cone:
Circle: z = 2
Ellipse: z = 2 + x/2
Parabola: z = 2 + x
Hyperbola: z = 2 + 2x
Rect. Hyp. x = 2
Two Lines: x = 0
One Line: z = x
Point: z = 0

PARALLEL AND PERPENDICLAR LINES
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Constants can be entered in equations and objects in Autograph. To illustrate the basic principles of vectors and lines in 2D:

1. Add a Vector line in the form

2. Add a point (a, b)
Add a point somewhere eg (3, 1), and add to that the vector [a, b]. Enter the line ax + by = c

Then use the constant controller to vary h, k, a, b, c to show that (a, b) is // to the vector line and ^ to the line ax + by = c. The equivalent in 3D is that the vector [a, b, c] is normal to the plane ax + by + cz = d

Enter points at A (-10,10) and B (22, 21) with axes edited as shown (note full control over axes start/finish and step values)

Enter velocity vector Va (15, 10) [red] and velocity vector Vb (–5, 0) [blue]

Reduce B to rest by adding –Vb (5, 0) [purple] to both B and A. Draw the vector sum of Va and –Vb to produce the motion of A as seen by B. So in each time unit, relative to B, A will ravel the length of Va – Vb.

You can create a straight line in Autograph:

- from two points
- in vector form:

- the intersection between 2 planes

You can create a plane in Autograph:

- from 3 points
- in vector form:

- implicit form: ax + by + cz + d = 0
[which is equivalent to r.n = d]

1. SKEW LINES
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Illustrated here are two skew lines, each created from two (moveable) points, and the “closest distance”. To illustrated that the closest distance is a mutually perpendicular direction, create a vector between each pair of points. Put a point on one end of the closest distance; select it and the two vectors and draw the cross product.

2. TWO PLANES AND INTERSECTION LINE
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Here two planes have been drawn and selected, and the intersection line drawn. With a point placed ON this line:
- select the point and one plane: draw normal unit vector
- select the point and the other plane: draw normal unit vector
- select the two vectors and the point, and draw the cross product. Note the direction of the cross product and the intersection line.

3. PARALLELOGRAM LAW IN 3D
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An illustration of the addition of 2 vectors to create the parallelogram law in 3D, together with the cross product.

4. THE INTERSECTION OF 3 PLANES
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1. SADDLE POINT
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Autograph file

The surface z = x² – y² has a saddle point, through which two straight lines can be drawn that always touch the surface.

These two lines can be entered parametrically,including '±':
x = t, y = ±t, z = 0

Use Ctrl-drag to move the camera in for a closer look

4. POLAR (SPHERICAL) - entered as r = f(θ, φ)
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Here, r = 3sin2θ has been plotted in 2D and in 3D. The 2D plot is equivalent to the intersection of the 3D with z= 0. The absence of φ in the equation means that a plane of the form x = ky will always intersect r = 3sin2θ in a circle. r = 1 plots a sphere.