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Vector algebra and trigonometric groundwork

Scope

  • Slides: pp. 52-57
  • Major topic folder: pslgs-dcels-vectors-and-geometric-primitives
  • Recording files touching this material: CS 564 - 01.23 1.2.txt, CS 564 - 01.30 3.1.txt
  • Goal of this file: You should be able to study this topic without reopening the slide deck.

Big picture

This is the algebra underneath every primitive. The course does not care about vector notation because it looks pretty; it cares because almost every geometric predicate becomes one subtraction and one determinant.

What you must know cold

  • Vector addition, subtraction, scalar multiplication, and translation view.
  • Direction as an angle/order concept, not just a picture.
  • Basic trigonometric facts used to compare directions or reason about left/right.

Core ideas and reasoning

  • Given points p and q, the vector q - p is the displacement from p to q.
  • Most primitives first translate so one point becomes the origin; then orientation and comparison become algebraic.
  • Trig is supporting material, but determinant-based comparisons are preferred because they avoid expensive angle computation.

Figures to actually look at

These are cropped from the main slide PDF. Do not skip them.

Figure from slide p. 55

Figure from slide p. 55

Figure from slide p. 57

Figure from slide p. 57

Slide-by-slide digestion

p. 52 - Vector algebra

  • An ordered pair (x, y) can be a point in the plane, or a vector.
  • Vector addition
  • Given vectors a = (xa, ya) and b = (xb, yb),
  • vector addition is defined as a + b = (xa + xb, ya + yb).
  • Geometrically, vectors a and b determine a parallelogram with
  • vertices 0, a, b, and a + b.
  • a + b

p. 53 - Scalar multiplication

  • Multiplication of vector b by a scalar (a real number) t.
  • Scalar multiplication is defined as tb = (txb, tyb).
  • The vector length is scaled by t.
  • If t < 0, the direction is reversed.
  • 2b
  • -b

p. 54 - Vector subtraction

  • Given vectors a = (xa, ya) and b = (xb, yb),
  • vector subtraction is defined as b - a = b + (-1)a,
  • carried out as b - a = (xb - xa, yb - ya).
  • b - a
  • Vector length
  • Length of vector a = (xa, ya) is defined as |a| = sqrt(xa
  • 2 + ya

p. 55 - Vector Translation

  • -a
  • b-a
  • Let a =op and b =oq. Then, b-a is a translation
  • of the vector pq at the origin o. Thus, two line
  • segments having same length and direction
  • are translates of each other and can be
  • identified with the same canonical line
  • segment originating at the origin o.

p. 56 - Vector direction

  • The direction of vector a is described by its polar angle θa,
  • the angle the vector makes with the positive x axis.
  • Measured in counterclockwise rotation,
  • starting at the positive x axis.
  • Values are in the range 0 ≤θa < 360.
  • θa
  • Given two vectors a and b, the angle between them θab
  • is measured counterclockwise starting at vector a.
  • θab

p. 57 - Trigonometry reminder

  • Definition of sine and cosine based on unit circle.
  • x = cos θ
  • y = sin θ
  • 0 < θ < 180
  • ⇒y > 0
  • ⇒sin θ > 0
  • 180 < θ < 360 ⇒y < 0
  • ⇒sin θ < 0
  • Unit circle x2 + y2 = 1

What you must be able to say or do in an exam

  • Give the precise definitions.
  • Distinguish similar notions cleanly.
  • Use the right primitive test or formula on a concrete example.

Complexity and performance facts

Primitive vector operations are constant time and are assumed cheap enough to use inside loops.

Common mistakes and danger points

  • Do not confuse a point with a vector until you have fixed the reference point.
  • Avoid actual angle computation when a sign test or determinant is enough.

Exam-style drills and answer skeletons

Definition drill

Question. Give the precise definitions and the most important consequences from vector algebra and trigonometric groundwork.

How to answer. A strong answer distinguishes similar objects and uses the course terminology exactly.

Recap

What you must know cold

  • Vector addition, subtraction, scalar multiplication, and translation view.
  • Direction as an angle/order concept, not just a picture.
  • Basic trigonometric facts used to compare directions or reason about left/right.

Core test / key idea

  • Given points p and q, the vector q - p is the displacement from p to q.
  • Most primitives first translate so one point becomes the origin; then orientation and comparison become algebraic.
  • Trig is supporting material, but determinant-based comparisons are preferred because they avoid expensive angle computation.

Complexity

  • Primitive vector operations are constant time and are assumed cheap enough to use inside loops.

Common mistakes / danger points

  • Do not confuse a point with a vector until you have fixed the reference point.
  • Avoid actual angle computation when a sign test or determinant is enough.

End-of-file summary

  • Vector addition, subtraction, scalar multiplication, and translation view.
  • Direction as an angle/order concept, not just a picture.
  • Basic trigonometric facts used to compare directions or reason about left/right.
  • Primitive vector operations are constant time and are assumed cheap enough to use inside loops.
  • Do not confuse a point with a vector until you have fixed the reference point.
  • Avoid actual angle computation when a sign test or determinant is enough.