Q BB notes, part a, Symmetry: After some thinking, and plugging in values for i, j, d, u, and v, I can sort of understand the equivalence. A All basis functions of the BB form on t in [0..1] are symmetric with respect to t=1/2. We used that property so we do not have to separately discuss what happens at t=0 and what happens at t=1. Q Parts b-g: confidence in my understanding. (I have a particularly hard time with the latter sections.) A To improve confidence, I recommend reviewing the sample test question and trying some of the questions in the hand-out. Note that the emphasis in any test is on the material covered in class and that you can take the notes along to the test. Q coordinate notes: I do not understand what the statement on page 2 means: the sum of lambda_i * p_i is allowable only if the sum of lambda_i is 1. ...Also, I thought you had said during lecture that it doesn't make sense to add points. A The context is that generically you cannot add (multiples) of points to points. We call this forming a linear combination. Only in the special case when the weights of such a linear combination of the points add up to 1 do you again have a point namely an average of the points. For example, 1/2 point A + 1/2 point B is well-defined (we used it to define de Casteljau's algorithm for t=1/2). Q I do not /really/ understand what affine space is, and how it is distinct from Euclidean space, as Euclidean space already has points and vectors (I thought). A (from http://en.wikipedia.org/wiki/Affine_space) In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. It can be thought of informally as a vector space where one has forgotten which point is the origin. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point that serves as an origin. less formally: The Cartesian coordinates with its preferrred point, called (0,0,0), and its orthogonal coordinates, for example (0,0,1)*(0,1,0)=0, is the special Eucledian setting. But another choice of origin and non-orthogonal coordinate axes are just as good for making the axioms of affine space work out. even less formally: don't worry too much about this (correct and ultimately important) distinction Q the example matrices for doing translation, rotation, and projection are 4x4. However, I thought embedding n-space into n+1-space was something related to homogenous coordinates and projective space. Are projective space and affine space not mutually exclusive types of spaces? A To visualize homogeneous n space, we can embed it into affine (even Eucledian) n+1 space. So it is not surprising that we can illustrate linear operations of homogeneous 3 space by linear operations on affine 4 space, namely multiplication with 4x4 matrices. ------------------------------------------------------------------ Q Projective space: I understand of the properties (points can be normalized, and all points that normalize to the same point are part of the same "equivalence class", things with the 4th coordinate as 0 are located at infinity). I read about it and various related topics on Wikipedia. A good practice that I very much encourage: free yourself from the (often outdated) textbooks -- this is what you have to do in real life. Q practical motivations absolutely aid in understanding. A That is why we do projects! Q Showing multiple examples with different control points (perhaps in extreme positions to create sharp curves or loops) would also help. A We have only limited time for such examples in class. Try the applets that I linked (we used them in class) or play with your project 1. Q For the BB form in particular, it might help to also show the power form for the equation, which everyone is familiar with, and then show how the BB form lends itself to easier intuitive interpretation. A We actually did that in class -- but converting polynomial representations to the power form is a bad habit.