Q: Can something be an eigenvalue of a 3 dimensional structure, and that structure being the graph of a complex polynomial?

A: Yes, it is possible for a complex number to be an eigenvalue of a 3-dimensional structure that is the graph of a complex polynomial.

To understand why, let's first define what we mean by the "graph of a complex polynomial". Given a complex polynomial P(z) of degree n, its graph is the set of all points (x,y,z) in three-dimensional space such that P(x+iy) = z. In other words, the graph of P is the set of all points (x,y,z) that satisfy the equation z = P(x+iy).

Now, let's consider the eigenvalues of this structure. An eigenvalue of a matrix A is a scalar λ such that there exists a non-zero vector v for which Av = λv. In our case, we can think of the graph of P as a matrix, where the entries are given by the values of P(x+iy) at each point (x,y,z) in three-dimensional space. Then, an eigenvalue of the graph of P would be a complex number λ such that there exists a non-zero vector v = (x,y,z) for which P(x+iy) = λz.

So, in summary, it is indeed possible for a complex number to be an eigenvalue of the graph of a complex polynomial in three-dimensional space.

Q: If I take a 2-dimensional view of a 3-dimensional situation, the "point" part of "point-of-view" and the "view" (which direction we are looking) together with the aperture, work to together to limit the 2-dimensional view. Is this correct?

A: Yes, that's correct! When you take a 2-dimensional view of a 3-dimensional situation, the point-of-view, view direction, and aperture work together to limit the 2-dimensional view.

The point-of-view refers to where you are standing or located relative to the 3-dimensional scene. Depending on your position, you may have a different view of the scene.

The view direction refers to the direction in which you are looking. This determines what part of the scene you see, and what part is hidden from view.

The aperture, also known as the field of view or angle of view, refers to the amount of the scene that is visible through your "viewing window". It determines how much of the scene is captured by the 2-dimensional view.

Together, these factors limit the 2-dimensional view of the 3-dimensional situation. The point-of-view and view direction determine what part of the scene is visible, while the aperture determines how much of the scene is visible.

Q: Posit a 3-dimensional surface and a point-of-view of it, which is external to it. The 2-dimensional view has a relationship to the 3-dimensional surface. Are there formal mathematical terms for this relationship?

A: Yes, there are formal mathematical terms to describe the relationship between a 3-dimensional surface and its 2-dimensional projection onto a plane, which is what you are describing.

The process of projecting a 3-dimensional surface onto a 2-dimensional plane is called "perspective projection". This is a mathematical process that involves mapping each point on the 3-dimensional surface to a corresponding point on the 2-dimensional plane.

The resulting 2-dimensional projection of the 3-dimensional surface is called a "perspective view", "perspective image", or "perspective drawing". This 2-dimensional projection preserves certain geometric relationships between points on the 3-dimensional surface, such as relative distances and angles, but it necessarily distorts other relationships, such as areas and shapes.

The mathematical relationship between the 3-dimensional surface and its 2-dimensional projection can be described using various mathematical concepts, such as linear algebra, differential geometry, and projective geometry. For example, in linear algebra, perspective projection can be described using a matrix transformation, while in projective geometry, it can be described using projective transformations.

In general, the relationship between a 3-dimensional surface and its 2-dimensional projection is a complex and fascinating area of mathematics that has many practical applications, including computer graphics, architectural drawing, and industrial design.