How to figure curves?

For everyone that subscribes to this thread, I want a dollar.

 

no way - you didn't start it!

 

Poor d reese. He will never get the answer he's looking for with all these jokers!

 

nor will he without a complete question - Advance Curve, Power Curve, Torque Curve, Tubing Curve, French Curve, Road Curve, Body Line Curve ad infinitum

 

Personally, I feel inspired by curves.
There are many mathematical formulas to figure curves.
look at these tires, lets say you wanted the to figure the circumference of the tire.
Measure the diameter and multiply by PIE or 3.1416
Simple!

 

After she takes your money wrecks your car has you thrown in jail then you might have time to check this out:

Given two bounded sets ​

A and B in Euclidean space Em, we define dB(A) to be​

d​

B​

(A) = sup​

a​

​

​

​

∈A​

d​

​

(a,B) (1)
where the distance from point a to set B is given by​

d​

​

(a,B) = inf​

b​

​

​

​

∈B​

d​

​

(a, b). (2)
The point distance d(a, b) may be any distance in Em, such as for example an p distance.
We refer to (2) as the minB calculation. In general dB(A) = dA(B). Letting Ballb() denote
the closed ball of radius centered at point b, the Minkowski -sausage of B, denoted by​

B​

​

, is the set​

B​

​

=​

​

b​

​

​

​

∈B​

Ball​

b().​

So ​

dB(A) is equal to the smallest such that A is contained in the -sausage of B.
Sets A and B could be the graphs of curves. Then, in 2-dimensional space, A =​

{​

​

(x(t), y(t)) : t ∈ [0, 1]} for some parameterized curve γ : t ∈ [0, 1] → (x(t), y(t)). Similarly
for B.
The Hausdorff distance between A and B will be taken as​

h​

​

(A,B) = dB(A) + dA(B).​

This distance between 2-dimensional sets is important in image processing in which the sets
are pixelized objects residing in a grid of ​

M ×N pixels or cells. Two objects A and B in a
black and white image are identical iff the Hausdorff distance between them is 0. Further,
if one object is the translate of the other by a distance t, B = A + t, then h(A,B) = 2t.
Letting |A| denote the number of pixels, or cardinality, of A, a straightforward computational
implementation of (1) calculates dB(A), and hence h(A,B) also, in O(|A||B|) time.
We refer to this as the Direct algorithm. However, it is possible to calculate h(A,B) for
discretized binary sets in time proportional to the frame size, MN, in two dimensions (see
(Shonkwiler, 1990) for the case of 1 point distances and (Shonkwiler, 1991) for the general​

​

p ​

case). We refer to these (collectively) as the Field algorithm, since the main idea is to
step from one cell to the next over the pixel grid.
Now suppose sets A and B are (discretized) curves. Their Hausdorff distance may be
computed by the above mentioned algorithms, but neither is efficient. In particular the
Field algorithm is inefficient since a curve is generally a sparse subset of the complete pixel
grid. In fact, if both A and B have Hausdorff dimension 1, then O(|A||B|) = O(MN). But
for such 1-dimensional curves, one would think that an algorithm linear in their arc-lengths​

|​

A| ​

+ |B| should be possible.
In this paper, we give a new algorithm for calculating (1) with an average complexity
of
log(max(M,N))(|A|+ |B|) (3)
2
when an p point metric with p = 1 and p = ∞ is used. (There are special configurations
of the sets for which (3) is violated when the point metric is 1 or ∞.) We refer to it as the​

Scaling algorithm​

​

, for its main idea is to refine an approximation of the distance h(A,B)
by rescaling the resolution and doing the direct calculation for only a small subset of pairs​

a ​

∈ A ​

and b ∈ B, which we call bridges. Unlike the Field algorithm, whose running time is
independent of A and B, the Scaling algorithm’s running time varies with the number of
points in the sets, as estimate (3) dictates. For curves, or more generally, sparse subsets,
and certain other subsets with widely separated points, the Scaling algorithm is faster. The
algorithm works, with only minor changes, for the various p point metrics. The Scaling
algorithm may be adapted to any space dimension; its advantage for sparse sets increases
with dimension. Finally, besides calculating the distance (1), the algorithm also produces
the points on the curves at which this distance is achieved.
In the next section, we present the Scaling algorithm in detail for two dimensions, but
here we outline the main ideas. For the purposes of computation, we assume the discretized
curves A and B live in an N × N square of cells, where N is a power of 2, N = 2R. This
may entail embedding the given frame in a larger one. The algorithm proceeds in stages​

r ​

​

= 0, 1, . . ., R and begins with the entire space containing A and B as one large block.
Processing at a given stage consists of these steps:
1 curve rescaling;
2 bridge updating;
3 minB (local) pruning, i.e. rescaling the d(a,B) candidates;
4 maxd (global) pruning, i.e. rescaling the dB(A) candidates.
Beginning with the crudest resolution possible, a single 1× 1 block, processing works
in stages toward the highest resolution. In this way, many portions of both curves are identified
early on as non-contenders in figuring into the calculation of dB(A). The remaining
contenders are kept for further processing, the bridges, consisting of candidate pairs of
blocks, or bridgeheads, from the representatives of A and B at the given resolution. To
carry out the updating, the blocks serving as bridgeheads are refined according to their
respective curves; this is step 1. Next, step 2: from the refined bridgeheads, all possibilities
for new bridges are considered, but many of them can be eliminated as contenders.
In step 3, the list is pruned by invoking equation (2) for each A bridgehead; this is a local
comparison. In step 4, a global pruning is carried out based on equation (1), and every
bridge length is compared with the current maximum.
Following the detailed explanation of these steps, we give some intermediate results​

of the algorithm and the number of bridges at each stage, in a typical application.


YOU WERE TALKING ABOUT BRIDGES WEREN"T YOU

 

What Tires?

 

I like boobs! ^^^^ Sorry, brain melting...

 

Ellipses can be confusing butt fun.
I use this alot, it help you figure how big of a piece to start with to get the compound curves.
Approximations to ellipses An ellipse of low eccentricity can be represented reasonably accurately by a circle with its centre offset. With the exception of Mercury, all the planets have an orbit whose minor axis differs from the major axis by less than half of one percent. To draw the orbit with a pair of compasses the centre of the circle should be offset from the focus by an amount equal to the eccentricity multiplied by the radius.

Two perfect examples of elliptical curves gone compound.

 

Sir Jackie Stewart summed it up best: "Exit speed is far more important than entry speed."

 

I can't !

I already gave my dollar to the Young Lady in post # 20

 

Oh, there all close on the answer, I just happen to have a different angle!!!

 

What was the question again ?????????

 

lovely thread guys

 

She was a Pirates dream, sunken chest.

She was a carpenters daughter, flat as a board.

In figuring out sheet metal patterns Integral Calculus is handy for designing curves.

 

What he said....

 

Do I detect a note of sarcasm here?
shouldn't you be sanding and slinging bondo?

 

Thats how I figure them ,Whatever is pleasing to the eye..........

 

Wow one of the greatest knocks ive ever heard on here. Damn!!!

 

All good curve design involves the strategic use of H2o

 

God, I love dagmars!

 

divco13 where do I get a seasonticket?

 

I figure if they aint got any then they're probably someones uncle in someones aunts underwear .




.

 

Black stockin's and hi-heels...

 

bwaaahaahaaahaa!!!

 

Don't forget if the curves are to big you won't be able to get your hands around them.
And they will eat you out of house and home too.

 

mmmmmmmm pie!!

BWAHAHAHAHAHAHA!!!!!! BURN!