The good pseudo-half-circle-like curves that quadratic bezier curves can create

1

$begingroup$

Since quadratic bezier curves can only approximate circles, I'm wondering what they can actually do well. That is, the sort of half-circle-like shapes that quadratic curves can make. For instance, in the letter u, maybe it's no good at creating a half-circle, but it might be really good at creating the U shape in some other fashion, maybe with just 1 or two curves that has some x contour. Wondering what that curve looks like and what is definition or properties are.

circle bezier-curve

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asked Jan 23 at 2:56

Lance PollardLance Pollard

1,358926

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have a particular definition of "well", e.g., general visual similarity, minimum # of curves required to approximate, small maximum difference, minimal total squared differences, low total absolute differences, etc.?
  $endgroup$
  – John Omielan
  Jan 23 at 3:00
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Basically what they can do with the minimum amount of curves where you're not trying to force it into something it can't do accurately and easily (like a circle). Minimum data to get a U-like arc (not a parabola).
  $endgroup$
  – Lance Pollard
  Jan 23 at 3:02
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for the clarification.
  $endgroup$
  – John Omielan
  Jan 23 at 3:04
  

  
  
  
  

add a comment | 

1

$begingroup$

Since quadratic bezier curves can only approximate circles, I'm wondering what they can actually do well. That is, the sort of half-circle-like shapes that quadratic curves can make. For instance, in the letter u, maybe it's no good at creating a half-circle, but it might be really good at creating the U shape in some other fashion, maybe with just 1 or two curves that has some x contour. Wondering what that curve looks like and what is definition or properties are.

circle bezier-curve

share|cite|improve this question

asked Jan 23 at 2:56

Lance PollardLance Pollard

1,358926

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have a particular definition of "well", e.g., general visual similarity, minimum # of curves required to approximate, small maximum difference, minimal total squared differences, low total absolute differences, etc.?
  $endgroup$
  – John Omielan
  Jan 23 at 3:00
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Basically what they can do with the minimum amount of curves where you're not trying to force it into something it can't do accurately and easily (like a circle). Minimum data to get a U-like arc (not a parabola).
  $endgroup$
  – Lance Pollard
  Jan 23 at 3:02
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for the clarification.
  $endgroup$
  – John Omielan
  Jan 23 at 3:04
  

  
  
  
  

add a comment | 

1

1

1

$begingroup$

Since quadratic bezier curves can only approximate circles, I'm wondering what they can actually do well. That is, the sort of half-circle-like shapes that quadratic curves can make. For instance, in the letter u, maybe it's no good at creating a half-circle, but it might be really good at creating the U shape in some other fashion, maybe with just 1 or two curves that has some x contour. Wondering what that curve looks like and what is definition or properties are.

circle bezier-curve

share|cite|improve this question

asked Jan 23 at 2:56

Lance PollardLance Pollard

1,358926

$endgroup$

Since quadratic bezier curves can only approximate circles, I'm wondering what they can actually do well. That is, the sort of half-circle-like shapes that quadratic curves can make. For instance, in the letter u, maybe it's no good at creating a half-circle, but it might be really good at creating the U shape in some other fashion, maybe with just 1 or two curves that has some x contour. Wondering what that curve looks like and what is definition or properties are.

circle bezier-curve

circle bezier-curve

share|cite|improve this question

asked Jan 23 at 2:56

Lance PollardLance Pollard

1,358926

share|cite|improve this question

asked Jan 23 at 2:56

Lance PollardLance Pollard

1,358926

share|cite|improve this question

share|cite|improve this question

asked Jan 23 at 2:56

Lance PollardLance Pollard

1,358926

asked Jan 23 at 2:56

Lance PollardLance Pollard

1,358926

asked Jan 23 at 2:56

Lance PollardLance Pollard

1,358926

1,358926

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have a particular definition of "well", e.g., general visual similarity, minimum # of curves required to approximate, small maximum difference, minimal total squared differences, low total absolute differences, etc.?
  $endgroup$
  – John Omielan
  Jan 23 at 3:00
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Basically what they can do with the minimum amount of curves where you're not trying to force it into something it can't do accurately and easily (like a circle). Minimum data to get a U-like arc (not a parabola).
  $endgroup$
  – Lance Pollard
  Jan 23 at 3:02
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for the clarification.
  $endgroup$
  – John Omielan
  Jan 23 at 3:04
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you have a particular definition of "well", e.g., general visual similarity, minimum # of curves required to approximate, small maximum difference, minimal total squared differences, low total absolute differences, etc.?
  $endgroup$
  – John Omielan
  Jan 23 at 3:00
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Basically what they can do with the minimum amount of curves where you're not trying to force it into something it can't do accurately and easily (like a circle). Minimum data to get a U-like arc (not a parabola).
  $endgroup$
  – Lance Pollard
  Jan 23 at 3:02
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for the clarification.
  $endgroup$
  – John Omielan
  Jan 23 at 3:04
  

  
  
  
  

$begingroup$
Do you have a particular definition of "well", e.g., general visual similarity, minimum # of curves required to approximate, small maximum difference, minimal total squared differences, low total absolute differences, etc.?
$endgroup$
– John Omielan
Jan 23 at 3:00

$begingroup$
Do you have a particular definition of "well", e.g., general visual similarity, minimum # of curves required to approximate, small maximum difference, minimal total squared differences, low total absolute differences, etc.?
$endgroup$
– John Omielan
Jan 23 at 3:00

$begingroup$
Basically what they can do with the minimum amount of curves where you're not trying to force it into something it can't do accurately and easily (like a circle). Minimum data to get a U-like arc (not a parabola).
$endgroup$
– Lance Pollard
Jan 23 at 3:02

$begingroup$
Basically what they can do with the minimum amount of curves where you're not trying to force it into something it can't do accurately and easily (like a circle). Minimum data to get a U-like arc (not a parabola).
$endgroup$
– Lance Pollard
Jan 23 at 3:02

$begingroup$
Thanks for the clarification.
$endgroup$
– John Omielan
Jan 23 at 3:04

$begingroup$
Thanks for the clarification.
$endgroup$
– John Omielan
Jan 23 at 3:04

add a comment | 

1 Answer
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I can think of a few good things about quadratic Bézier curves:

1. They can represent parabolas exactly. Cubic curves can do this too, of course, but not as economically.


2. It's very easy to do computations with them. For example, intersection with a straight line requires only solution of a quadratic equation.


3. They are easy to convert to implicit form, so it's easy to check whether a given point lies on the curve.


4. Their shapes are very predictable. If the control points are $A$, $B$, $C$, then the curve passes through the point $tfrac12bigl(B + tfrac12(A+C) bigr)$, and it's tangent at this point to the line joining $tfrac12(A+B)$ and $tfrac12(B+C)$. So, there is an obvious and direct relationship between the shape of the control polygon and the shape of the curve.


5. Quadratics never have inflexions, which is important in some design scenarios.

If you want to approximate some complex shape, then using quadratic pieces will require you to use more (and shorter) segments than when using cubics. But each quadratic segment will require less data than a cubic one. Compare TrueType fonts (quadratic segments) with Postscript ones (cubic segments) to get a sense of this.

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edited Feb 5 at 6:27

answered Jan 31 at 23:16

bubbabubba

30.7k33188

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add a comment | 

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1 Answer
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1

$begingroup$

I can think of a few good things about quadratic Bézier curves:

1. They can represent parabolas exactly. Cubic curves can do this too, of course, but not as economically.


2. It's very easy to do computations with them. For example, intersection with a straight line requires only solution of a quadratic equation.


3. They are easy to convert to implicit form, so it's easy to check whether a given point lies on the curve.


4. Their shapes are very predictable. If the control points are $A$, $B$, $C$, then the curve passes through the point $tfrac12bigl(B + tfrac12(A+C) bigr)$, and it's tangent at this point to the line joining $tfrac12(A+B)$ and $tfrac12(B+C)$. So, there is an obvious and direct relationship between the shape of the control polygon and the shape of the curve.


5. Quadratics never have inflexions, which is important in some design scenarios.

If you want to approximate some complex shape, then using quadratic pieces will require you to use more (and shorter) segments than when using cubics. But each quadratic segment will require less data than a cubic one. Compare TrueType fonts (quadratic segments) with Postscript ones (cubic segments) to get a sense of this.

share|cite|improve this answer

edited Feb 5 at 6:27

answered Jan 31 at 23:16

bubbabubba

30.7k33188

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add a comment | 

1

$begingroup$

I can think of a few good things about quadratic Bézier curves:

1. They can represent parabolas exactly. Cubic curves can do this too, of course, but not as economically.


2. It's very easy to do computations with them. For example, intersection with a straight line requires only solution of a quadratic equation.


3. They are easy to convert to implicit form, so it's easy to check whether a given point lies on the curve.


4. Their shapes are very predictable. If the control points are $A$, $B$, $C$, then the curve passes through the point $tfrac12bigl(B + tfrac12(A+C) bigr)$, and it's tangent at this point to the line joining $tfrac12(A+B)$ and $tfrac12(B+C)$. So, there is an obvious and direct relationship between the shape of the control polygon and the shape of the curve.


5. Quadratics never have inflexions, which is important in some design scenarios.

If you want to approximate some complex shape, then using quadratic pieces will require you to use more (and shorter) segments than when using cubics. But each quadratic segment will require less data than a cubic one. Compare TrueType fonts (quadratic segments) with Postscript ones (cubic segments) to get a sense of this.

share|cite|improve this answer

edited Feb 5 at 6:27

answered Jan 31 at 23:16

bubbabubba

30.7k33188

$endgroup$

add a comment | 

1

1

1

$begingroup$

I can think of a few good things about quadratic Bézier curves:

1. They can represent parabolas exactly. Cubic curves can do this too, of course, but not as economically.


2. It's very easy to do computations with them. For example, intersection with a straight line requires only solution of a quadratic equation.


3. They are easy to convert to implicit form, so it's easy to check whether a given point lies on the curve.


4. Their shapes are very predictable. If the control points are $A$, $B$, $C$, then the curve passes through the point $tfrac12bigl(B + tfrac12(A+C) bigr)$, and it's tangent at this point to the line joining $tfrac12(A+B)$ and $tfrac12(B+C)$. So, there is an obvious and direct relationship between the shape of the control polygon and the shape of the curve.


5. Quadratics never have inflexions, which is important in some design scenarios.

If you want to approximate some complex shape, then using quadratic pieces will require you to use more (and shorter) segments than when using cubics. But each quadratic segment will require less data than a cubic one. Compare TrueType fonts (quadratic segments) with Postscript ones (cubic segments) to get a sense of this.

share|cite|improve this answer

edited Feb 5 at 6:27

answered Jan 31 at 23:16

bubbabubba

30.7k33188

$endgroup$

I can think of a few good things about quadratic Bézier curves:

1. They can represent parabolas exactly. Cubic curves can do this too, of course, but not as economically.


2. It's very easy to do computations with them. For example, intersection with a straight line requires only solution of a quadratic equation.


3. They are easy to convert to implicit form, so it's easy to check whether a given point lies on the curve.


4. Their shapes are very predictable. If the control points are $A$, $B$, $C$, then the curve passes through the point $tfrac12bigl(B + tfrac12(A+C) bigr)$, and it's tangent at this point to the line joining $tfrac12(A+B)$ and $tfrac12(B+C)$. So, there is an obvious and direct relationship between the shape of the control polygon and the shape of the curve.


5. Quadratics never have inflexions, which is important in some design scenarios.

If you want to approximate some complex shape, then using quadratic pieces will require you to use more (and shorter) segments than when using cubics. But each quadratic segment will require less data than a cubic one. Compare TrueType fonts (quadratic segments) with Postscript ones (cubic segments) to get a sense of this.

share|cite|improve this answer

edited Feb 5 at 6:27

answered Jan 31 at 23:16

bubbabubba

30.7k33188

share|cite|improve this answer

share|cite|improve this answer

edited Feb 5 at 6:27

edited Feb 5 at 6:27

edited Feb 5 at 6:27

answered Jan 31 at 23:16

bubbabubba

30.7k33188

answered Jan 31 at 23:16

bubbabubba

30.7k33188

answered Jan 31 at 23:16

bubbabubba

30.7k33188

30.7k33188

add a comment | 

add a comment | 

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