Mtg 9/23: Mon-07-Oct-2024

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Geometric Objects and Transformations

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  • Hello, Monday, October, 7, You.
  • So continue with chapter four. I want to bring to your attention
  • a couple of events on Friday. So there's the lunch
  • in Rick, 219,
  • the lunch theater. And
  • then there's also the seminar at the reception to follow.
  • So that was an education auditorium, but it's been moved
  • to Rick, 219,
  • so that makes easier for for the hospitality folks to arrange
  • things and coordinate amongst different activities going on
  • the day that day. So the reception is in the lab calf,
  • who calls it the lab calf, besides me or lab Cafe anyway,
  • that's
  • very close to The record. Key, One, Nine, I,
  • that's F, not A T.
  • So I'd like your input
  • on the Judea, Could you close the door? I
  • thank you. Now we'll see Whether it helps or not you
  • so inspired by a 16 by 16 favicon. Anyone familiar with
  • favicons, that's the little picture that comes up
  • beside the website Like
  • the F for Facebook there and so on. So.
  • So I'm not
  • going to beat this or play with this point, but I
  • so I'm thinking, we're thinking about doing u
  • r, CS, I
  • so then we have to decide which cupcakes to ice with the
  • foreground color and which ones to ice With the background color
  • to bring To make this A display so
  • so 16 and 16 being with more characters and
  • more characters,
  • eight by eight.
  • Screen, so you can see that I have an app here to do the IBM
  • BIOS font to
  • is this related to class at All closely enough that I could
  • bring it up? I
  • more food.
  • Okay, so?
  • Let's look At last Day I
  • so
  • We use matrices. I
  • so matrices can be used
  • to specify transformations on points and
  • vectors.
  • If we want to do a three dimensional transformation, is
  • it sufficient To have a three dimensional Matrix, three value
  • matrix, if
  • So we have a three by three matrix to represent the
  • rotation, scaling and sharing that we want to apply to our
  • model, but we don't have a way to express the translation, So
  • you
  • need to have a translation Vector and
  • so we have point P, we transform it, and we need to do the
  • addition as well. So what can we make that simpler? The idea of
  • transforming
  • our geometric primitives And
  • you? Yeah, so
  • changing His coordinates and
  • bless you. I
  • scale and shear. I,
  • so
  • your arms and lights and
  • like This
  • vector, a column vector and
  • a one by four array or one by four matrix and a
  • vector is
  • so we have the extra component, w, x, y, z and w. So w is one
  • for points and zero
  • or vectors. So
  • how do we make that distinction? Because we think of a vector as
  • the difference between two points.
  • So if
  • we're subtracting 2.1 minus one is Zero And
  • so let's talk a bit about how to identify the axis On which
  • something is spinning. I
  • So to begin with our spinning on the x axis and
  • so let's talk about how this could be made more clear,
  • because What we're doing, if we choose to rotate
  • on y
  • we're left with the value that was set for the rotation on the
  • x axis, and now we're incrementing the y axis,
  • and then We can spin on the Z axis as well. And
  • so how do we get back to
  • our Starting position and
  • Okay, so what? What can we do to improve this display
  • so it's clear Where the
  • axis is?
  • Okay, Let's Do
  • So Any ideas I
  • see it on the what If we do Something like I
  • so Do from the origin i
  • And then We'll Say, I
  • Comments, questions. Could
  • you do an example of some of the teachers algorithms for like
  • tables? Let's say just like a Q over, like the y axis on just
  • really some of the math so we can get Like that Central
  • Understanding. Okay,
  • let's finish This. Okay,
  • so here we can see in the vertex shader the code for the
  • different rotations.
  • So remember, they're column major. So we Bring the matrix.
  • So
  • RX Is I
  • it's going to rotate it with the origin. So
  • let's see what's an angle we could rotate by
  • about 90 degrees if
  • cosine and
  • so cosine of 90 degrees zero. Sine of 90 degrees is one
  • column,
  • zero and
  • minus one and five cosine of 90 is 00001,
  • okay, let the fiber routine.
  • Pick a number between one and 10,
  • seven,
  • Next two
  • and one more, five, one So.
  • So we get seven and
  • then it's minus five and
  • rotate around the z the x axis by 90 degrees.
  • So We have plane
  • set. 252,
  • and we rotate by 90 degrees. We have a
  • we have two in the z direction,
  • then minus
  • 5y direction and
  • seven. So we're staying put at seven.
  • We rotate between
  • the let's look at it from the x axis here. Let's put x here,
  • y and z.
  • So we're at x to the seven and
  • two and five, and Then
  • we go to
  • The I needed over explaining? It
  • does that make sense? So we're switching rotating around the x
  • axis.
  • So we start with five two here, and what's minus Five? Two.
  • So we're switching places. In that case, we're
  • okay, So let's write a sheets for y and sand, and then we'll
  • multiply them together. So
  • so CO, sign,
  • minus sign, first column, I,
  • okay, so what do we notice
  • about the columns or the axis about which we're rotating? So
  • we rotate around x, the first column
  • is x,
  • that's part of the identity matrix. And for
  • y, we have the same thing. And then from S, Y and spy, you
  • any questions back there? Okay. I
  • so what is the z column going to look like for let's write
  • it as rooting around z you can
  • tell me what the said column will be for The Z rotation
  • matrix.
  • Start at zero, next zero. Yeah.
  • Zero. Next,
  • one, zero.
  • Do that, right? I pronounced.
  • Okay, so how about for me? Let me see you.
  • So I believe that is square root of two over two,
  • which happens to be point squares, point 7271, So One, you
  • rotating around the Y axis which point is going to remain
  • constant under rotation.
  • Okay, so, right, seven to a seven times Seven
  • plus
  • nine, minus 7271, seven
  • times, Five So.
  • So let's say, instance,
  • eight line four and four. This
  • One is minus four by nine plus three. Five, so
  • that's minus 1.4 so That's 8.4
  • so two axis look at x In the
  • net, we have seven And five,
  • and rotate by 45 degrees. Approximately
  • 8.4 minus 1.4
  • I should have just done a 90 degree angle again. I
  • so let's do 90 here again so you
  • can check my math here.
  • Cosine 90 is zero. Sine of 90 is 1000,
  • minus one And
  • so we have minus two And then
  • five and one and
  • Okay, Let's make this minus 90 here. I'm
  • okay, so let's Do your Patience And
  • rotate by x first and rotate by y, rotate by Z.
  • That's what we're doing in the shade right here. So
  • this is really good anyway.
  • So so rotate my x Is unserved and
  • and Zero.
  • I throughout the time where we
  • i Well, double check the work here and see what we get for the
  • answer To transform
  • so describing what I'm asking
  • for, I prescribing project.
  • Anyway, thank you, everyone.
  • Bye. Shop everywhere. Oh,
  • Bye. Shop everywhere. Oh,

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