apma unit 1

Implicit surfaces in 3D and variable roles

• Surfaces in 3D are given by equations in three variables (x, y, z). They are sets of points that satisfy the equation, not fixed numerical substitutions for x, y, or z.
• If you think in spherical coordinates (ρ, φ, θ), a surface can be described by a relation among these variables. For a sphere centered at the origin, ρ is constant (ρ = R). If the sphere is not centered at the origin, ρ is not constant in direction; ρ becomes a direction-dependent quantity, and the radius relates to the sphere’s center rather than the origin.
• In discussions of surfaces, you often end with an equation that involves three variables with no substitutions: for a sphere of radius R centered at the origin, you typically end with an equation like ρ = R in appropriate coordinates or, in Cartesian form, x, y, z as variables.
• There can be a question about whether to show the derivation; sometimes the simplest final form is accepted (e.g., ρ = 6 for a sphere of radius 6). The presenter notes that there’s “no work to do” to obtain the surface equation in the standard form, though you could derive it via substitution and trig identities if you wanted.
• Example mentioned: from the sphere equation x^2 + y^2 + z^2 = 36, one could manipulate to get ρ = 6, but that would be unnecessarily roundabout.

Sphere, planes, and traces

• A plane parallel to a coordinate plane is called a fundamental plane. The coordinate-aligned planes have equations of the form:
  • x = a (parallel to the yz-plane)
  • y = b (parallel to the xz-plane)
  • z = c (parallel to the xy-plane)
• Intersecting a fundamental plane with a sphere centered at the origin yields a circle (the trace).
• Traces are two-dimensional slices of a three-dimensional surface along coordinate planes, used to build intuition about the 3D shape.
• When you slice a sphere with a fundamental plane, you always get a circle; for more general quadric surfaces, traces are typically ellipses or other conic sections depending on the slice.
• Slices along x = a, y = b, or z = c help visualize the 3D surface as a collection of 2D shapes.

Sphere: standard form and normalization

• The equation of a sphere centered at the origin with radius R:
  • $x^2 + y^2 + z^2 = R^2.$
• If R^2 = 36, dividing both sides by 36 yields the standardized form with a unit right-hand side:
  • $\frac{x^2}{36} + \frac{y^2}{36} + \frac{z^2}{36} = 1.$
• The right-hand side is 1, and the three squared terms have identical coefficients when written in this normalized form (coefficients on x^2, y^2, z^2 are the same).
• In standard 3D form that emphasizes the radii along each axis, you can write:
  • $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,$ where for a sphere a = b = c = R.
• If you keep the right-hand side as 1 but change coefficients, you move away from a sphere toward ellipsoids.

Ellipsoid: coefficients and traces

• Ellipsoids are the 3D analogs of ellipses (spheres are a special case of ellipsoids).
• Example ellipsoid equation (as discussed):
  • $x^2 + \frac{1}{2} y^2 + 2 z^2 = 1.$
  • This is an ellipsoid with different extents along the x, y, and z axes due to the different coefficients.
• If you plot this implicit surface, you get an ellipsoid that is stretched or compressed along different axes depending on the coefficients (e.g., it may be longer in x, shorter in y, etc.).
• Traces of an ellipsoid on the coordinate planes are generally ellipses; sometimes you might see a circle if the coefficients align to give equal extents in two directions (
  e.g., if the cross-section along a particular axis yields equal radii).
• For this ellipsoid example, a plane like y = 0.4 would intersect the surface in an ellipse shape outline, illustrating how traces reflect the 3D geometry.

Hyperboloids: one sheet and two sheets

• Hyperboloids are the 3D analogs of hyperbolas, with two common types discussed:
  • Hyperboloid of one sheet: formed when two terms are positive and one is negative on the left-hand side, equated to 1. Example pattern (any permutation of axes):
  • $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1.$
  • The surface is single-connected (one piece) and can be wider along certain directions depending on coefficients.
  • Hyperboloid of two sheets: formed when two terms are negative and one is positive on the left-hand side, equated to 1. Example pattern:
  • $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$
  • The surface consists of two separate pieces (two sheets), separated along the axis with the positive term.
• Orientation depends on which axis carries the negative sign; switching the negative sign to a different axis yields the same qualitative shape, just oriented differently (e.g., the two sheets could be along the z-axis or along the x-axis, etc.).
• If you have three negative coefficients:
  • $-\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1.$
  • There are no real points that satisfy this (sums of three negative squares cannot equal +1). This is the empty set, not a surface.
• A quick contrast to 2D: in R^2 you cannot have two negative squares equal to a positive constant; negative x^2 - y^2 = 1 has no real solutions, whereas 3D allows hyperboloids with one or two sheets. A triple-negative equation yields the empty set rather than a new surface.

Traces, intuition, and computational notes

• Traces are useful for building intuition about 3D surfaces by examining 2D intersections with coordinate planes.
• Common trace observations:
  • Sphere traces on coordinate planes are circles (as a consequence of symmetry).
  • Ellipsoid traces are generally ellipses; specific coefficient choices can yield circles in some traces.
  • Hyperboloid traces can be ellipses or hyperbolas in the 2D plane depending on the slice, and their 3D shape is understood through these traces.
• Implicit surface plotting tools can render these shapes; implicit surface plotting helps visualize how changing coefficients affects the geometry.
• The discussion emphasizes thinking in 2D slices (traces) to understand 3D quadric surfaces rather than relying solely on full 3D mental models.

Quick formulas and recap

• Sphere (center at origin):
  • $x^2 + y^2 + z^2 = R^2.$
  • If R^2 = 36: $x^2 + y^2 + z^2 = 36.$
  • Normalized form with RHS = 1: $\frac{x^2}{R^2} + \frac{y^2}{R^2} + \frac{z^2}{R^2} = 1.$
• Ellipsoid (general form):
  • $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1, \quad a,b,c>0.$
  • Example: $x^2 + \frac{1}{2} y^2 + 2 z^2 = 1.$
• Hyperboloid of one sheet (two positives, one negative):
  • $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1.$
• Hyperboloid of two sheets (two negatives, one positive):
  • $-\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$
• Empty set example (no real points):
  • $-\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1.$
• Important takeaway: the signs of the squared terms and the right-hand side determine the type of quadric surface (sphere, ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, or empty set).

Class logistics and next steps (as discussed in the lecture)

• Hard copies are available for those who want them; otherwise, students can download materials from Canvas on Wednesday.
• The next topics planned: 1.05 tomorrow; 1.06 may be postponed. The last part of the class typically includes a worksheet to be completed in class and submitted through Canvas, with two opportunities to revise after grading.
• Collaboration is allowed for discussion, but individual submissions are required.
• The TA (Laura) will assist during the class; students can raise hands for help and obtain hard copies if desired.
• The material emphasizes using traces and slices to understand 3D quadric surfaces and building intuition through 2D cross-sections.