Computer Graphics (CS4300) 2011S: Lecture 11

Today

introduction to curves in 2D
parametric curves
implicit curves
local curve properties

What is a Curve?

intuitively, a curve in 2D is “what you get when you draw with a pencil on paper without lifting your hand”
how can we formalize that intuition?
observe that curve is effectively a one-dimensional object (i.e., you can know “where you are” along a curve given only one real number), but it is embedded in a two-dimensional space
this motivates the idea of a continuous mapping from some connected interval of the one-dimensional real line , i.e. , to the two dimensional space
- each 1D point in , i.e. , is mapped to a point in
- the collection of all points is called the image of under the mapping
- the mapping is continuous iff the image is connected, i.e. if there are no “breaks” in the image
- we will call such an image a curve
- it’s also possible to define a curve to be the mapping itself
we will allow either or both to be infinite, i.e., we will allow curves that “never end”
also we will allow to be either closed (includes both endpoints), open (includes neither endpoint), or half-open (includes exactly one endpoint)

Parametric Curves

the above development motivates an idea for defining curves given two pieces of information
- a specific parameter interval
- a vector-valued continuous function
then, for finite ,
- the “first” point on the curve is
- the “last” point is , and
- the point “moves along” the curve as varies from to
- the continuity of ensures that the image , i.e. the set of all points mapped from , is continuous
if or is infinite, or if is not closed, the only difference is that there simply is not any “first” or “last” point
in general, will not be the actual Euclidean distance traveled from to along the curve
- however, if we happen to know the total length of the curve (i.e. the distance you’d travel if you had to “drive” from to along the curve, not the distance “as the crow flies”), we can re-scale the parametrization and define (assuming ), with , i.e.
- this is called an arc length parametrization because is the actual distance traveled along the curve from to
- any curve can be arc length parametrized, but to do so essentially requires that the length of the curve be calculated
- because we have given so little restriction on the function , in general calculating the length of the curve may require computing an integral, and there may be no “closed form” solution (i.e. you could have to numerically integrate to get an approximate answer for )
example: line segment
- we have already studied the parametric form for a line segment
- given the start and end of the segment as vectors ,
- we take and
- if we define then we can simplify to
- if we want an arc length parametrization, we can use with
example: line
- a line is different from a line segment: a segment has finite extent, but a line goes on to infinity in both directions
- implication: in all cases the parameter interval will be unbounded, i.e.
- recall that a line may be given in different forms
- two points on the line
  - is a parametrization of the line
- one point on the line and a vector pointing along the line
  - is a parametrization of the line
- one point on the line and a unit vector pointing along the line
  - is an arc-length parametrization of the line
- the unit normal of the line and a distance from the origin to the line
  - is a point on the line
  - is a unit vector along the line
  - proceed as above with and
- an orientation of the unit normal and a distance from the origin to the line
  - same as above, with
- an equation of the form
  - we’ll discuss this later!
example: ray
- a ray is a mix between a segment and a line: it’s only infinite in one direction
- assume we’re given the start point of the ray and a unit direction vector
- and is an arc-length parametrization of the ray
example: circle
- given center and radius
- and
- if we want an arc-length parametrization, ,
example: ellipse
- given center , horizontal radius , and vertical radius
- and
- this produces an axis-aligned ellipse
- for a rotated ellipse, one method is to apply a rotation matrix to

Implicit Curves

when we studied lines, we learned that an alternative way to describe a line is to give an equation of the form
this is called an implicit equation for the line, because unlike the above parametric forms, it does not give a direct procedure for computing a point on the line given some parameter
but it does imply all points on the line: a point is on the line exactly when the equation holds, i.e. when the expression evaluates to zero
one reason parametric forms are useful is that the value of the form for an arbitrary point can usually be taken as a measure of the distance from that point to the curve
there are also implicit forms for some other common curves
unlike the parametric approach, it is not directly possible to represent a curve with finite endpoints in implicit form
we will review the implicit form for a line, and then we will give implicit forms for two other common closed curves, i.e., curves without endpoints: circles and ellipses
example: line
- we have already covered this in detail
- one nice way is to consider we are given a unit normal and distance from the origin to the line (i.e. in the direction )
- any point in the plane on the line can be decomposed into a component in the direction and a remaining component parallel to the line
- i.e.
- we can compute directly as
- that is, the scalar is the length of , and is its direction
- (in this case it is not necessary to actually calculate , but if we wanted it, observe that implies )
- intuition: the length of will equal iff is on the line
- thus a valid implicit equation for the line is or, rearranging,
- this is just a fancy way to write , i.e. , , .
- as we have seen before, a similar equation works even given a normal vector that may not be unit length, and any set of constants can be interpreted as giving a normal vector and scaled distance , as long as are not both zero
- but given a unit normal, the value of the expression is the actual Euclidean distance from to the line (otherwise, it will be a scaled distance)
- in any case, the sign of tells us what side of the line is on:
  - if positive, is on the “left” of the line, where that is defined to be the side that the normal vector “points towards”
  - if negative, is to the “right” of the line
  - and of course is on the line exactly when is zero
example: circle
- intuition: the Euclidean distance from a point on the circle to the center of the circle must equal the circle radius
- so given the circle center and radius , a point is on the circle exactly when
- rearranging gives
- the expression will be negative for all points strictly inside the circle, and positive for all points strictly outside
- in fact, it is easy to see that is the Euclidean distance from to the nearest point on the circle
example: ellipse
- given center , horizontal radius , and vertical radius , apply a scale matrix to the equation for a unit-radius circle:
- this is an implicit equation for an axis-aligned ellipse
- this can be rewritten in component form as , and if , we get an equation that may be familiar to you:
- value will still be negative for strictly inside ellipse, positive for strictly outside, but does not give the actual Euclidean distance from to the nearest point on the ellipse

Local Curve Properties

a local property of a curve is something we can say about the curve at a given point on the curve, only considering the local neighborhood around that point
for example,
- what direction is the curve “facing”, i.e. what is the tangent to the curve?
- how fast is the curve “bending”, i.e., what is the curvature here?
- are we at an endpoint, i.e. is the curve continuous at this point?
contrast these with the following global properties:
- does the curve have any endpoints?
- does the curve ever cross itself?
- if the curve is closed, what area does it contain?
we will now develop methods for finding some local properties of a parametric curve
we can always find a local tangent vector to a parametric curve at a given point by taking the vector derivative of the curve
- that is, if , then
- the length of this vector is the speed of the curve at (or just “at ”), and the direction is the “forward” direction of the curve at that point
- it turns out that if the curve is arc-length parametrized then its speed will always be 1
- for a line given as , is just
- for an arc-length parametrized line given as , is just , a unit vector as expected
- for a circle given as with , , so the speed is always and is always tangent to the circle, pointing CCW
- for an arc-length parametrized circle with , , same as before but now with unit magnitude (since )
- for an ellipse given as with , , so the speed varies but still always tangent to the ellipse and points CCW
we can similarly always find a local normal vector to a parametric curve at a given point by taking the vector second derivative of the curve
- that is, if , then
- when the curve is arc-length parametrized, is called the curvature at (or just at )
- for an arc-length parametrized line given as , is , i.e. the line has no curvature
- for an arc-length parametrized circle with , , so the curvature is always , and always points inwards
- we haven’t presented an arc-length parametrized ellipse, but for an ellipse given as with , , so still at least always points inwards
a parametric curve has local position continuity (or simply “is continuous”) at iff the limit of is the same approaching from either direction
- this is just a precise way to say that the curve can be “drawn without lifting your pencil”
- this is also referred to as continuity, for reasons we will see shortly
- actually, by the way we defined a “curve”, every curve has continuity
- but sometimes, in particular when we talk about assembling a piecewise curve from individual parts, we need to be aware of continuity
similarly, a parametric curve has local tangent continuity at iff the limit of the first derivative is the same approaching from either direction
- intuitively, this means the curve does not have a sharp “kink” at
- this is also called continuity
- again, this is particularly useful when dealing with the joining of piecewise curves
- note that if the curve is not arc-length parametrized then it’s possible for the tangent vectors approaching from either direction to have the same (or opposing) direction but different magnitudes
  - technically our definition, which is based on parametric continuity, would say that the curve is not continuous in this case
  - if we don’t want to care about this, we can take a slightly modified definition of geometric continuity where we only care that the tangent vectors have the same direction
finally, a parametric curve has local curvature continuity at iff the limit of the second derivative is the same approaching from either direction
- intuitively, this means that the curve “bends in a consistent direction” at
- this is also called continuity (see the pattern?)
continuity is similarly defined for any integer
a curve is smooth if it is continuous for all
- though sometimes people use the word smooth to mean just that there is some such that the curve is continuous for all
- we can also write in that case that the curve is “ smooth”

Next Time

reading on website
polynomial curves in 2D
interpolating polynomials in 2D
interpolating splines in 2D