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| A '''Bézier triangle''' is a special type of [[Bézier surface]], which is created by (linear, quadratic, cubic or higher degree) interpolation of control points.
| | I gained a lot of fat when I was pregnant with my son, and my body happened to shop simply regarding every ounce left over for the next four years. It didn't feel wise understanding I hadn't loss anything, plus I'd had enough of feeling fat all the time. After seeing how heavy I actually looked inside my brother's wedding images, I decided to lose fat by setting a New Year's resolution!<br><br>Weight reduction is a main issue for nearly all of the folks considering it causes wellness risks. So should you join CAL system, a problem is solved easily and securely. During the course of the system, you are able to have unlimited access to the physician and health staff. You can see fat reduction clinics where the doctor will guide we at every step accurately. You can reach know which the rate of fat reduction change inside the body in an ongoing basis. Your weekly follow-up to the clinic can give we accurate information regarding your progress. The doctor might guide you proper exercises or when you want to have fast fat reduction, then the doctor might guide we thus by providing you with certain medicines.<br><br>Total body fat percentage consists of fat imperative to the metabolism in addition to stored fat. Essential fat is indispensable in terms of sustaining life and reproductive functions of the body. The imperative percentage is better for women than which for guys considering their bodies have to deal with the demands of childbearing plus additional hormonal functions. According to the accepted norms, necessary fat is 2-5% for men plus 10-13% for women. The minimal total body fat percentage suggested by healthcare experts is much more than this imperative fat percentage. Storage fat pertains to the fat accumulation in adipose tissue. Some of the adipose tissue is imperative for safeguarding internal organs in the chest and abdomen plus it is slightly high for females than for men.<br><br>The BMI index is broken down into four leading categories. Every category is represents a level of morbidity plus mortality data or wellness risk. If the [http://safedietplansforwomen.com/bmi-calculator bmi calculator] reads lower than 18.5 you're considered underweight, if it reads between 18.5 to 25 you're at an optimal weight; if the bmi calculator reads 25 to 30 your are obese, if it reads 30 or above than we are considered overweight (over 40 morbidly obese). Patients that score above a 30 are a major concern of the healthcare profession. Studies have shown which being obese may have an impact on the total health. These same studies show how weight is shortening the average existence span.<br><br>This is the case with a person i learn. She's 63, 5'9" plus really close to 300 lbs. While she can walk, in a limited way, her efforts to lose weight are just not working. She has alternative problems also like CHF plus osteoporosis. Her target weight is around 180, thus thats over 100 lbs she must loose. I devised this program for her!<br><br>Another limitation is that age is not considered when calculating BMI. So for elderly people, even if the BMI indication is regular, the truth remains which many muscles have been lost due to age.<br><br>These formulas aren't actual, nevertheless they provide fairly exact approximations. This usually give we the ability to measure the body fat plus have a better idea of how healthy you are. Use this info perfectly, plus keep up the wise work! |
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| ==Cubic Bézier triangle==
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| [[File:Bezier triangle.png|thumb|180px|An example Bézier triangle with control points marked]]
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| A '''cubic Bézier triangle''' is a [[surface]] with the equation
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| :<math>\begin{align}
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| p(s, t, u) = (\alpha s+\beta t+\gamma u)^3 =&
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| \beta^3\ t^3 + 3\ \alpha\beta^2\ st^2 + 3\ \beta^2\gamma\ t^2 u + \\
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| &3\ \alpha^2\beta\ s^2 t + 6\ \alpha\beta\gamma\ stu + 3\ \beta\gamma^2\ tu^2 + \\
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| &\alpha^3\ s^3+ 3\ \alpha^2\gamma\ s^2 u + 3\ \alpha\gamma^2\ su^2 + \gamma^3\ u^3
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| \end{align}</math>
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| where α<sup>3</sup>, β<sup>3</sup>, γ<sup>3</sup>, α<sup>2</sup>β, αβ<sup>2</sup>, β<sup>2</sup>γ, βγ<sup>2</sup>, αγ<sup>2</sup>, α<sup>2</sup>γ and αβγ are the [[Control point (mathematics)|control point]]s of the triangle and s, t, u (with 0 ≤ s, t, u ≤ 1 and s+t+u=1) the [[Barycentric coordinates (mathematics)|barycentric coordinates]] inside the triangle.<ref>{{citation|url=http://www.math.ubc.ca/~cass/courses/m308-03b/projects-03b/drader/main.htm|title=3D Surface Rendering in Postscript}}</ref>
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| <ref>{{citation | |
| |title=Curves and surfaces for computer-aided geometric design
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| |first=Gerald
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| |last=Farin
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| |publisher=[[Academic Press]] Science & Technology Books
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| |year=2002
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| |isbn=978-1-55860-737-8
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| |edition=5
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| }}</ref>
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| The corners of the triangle are the points α<sup>3</sup>, β<sup>3</sup> and γ<sup>3</sup>. The edges of the triangle are themselves [[Bézier curve]]s, with the same control points as the Bézier triangle.
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| By removing the γu term, a regular Bézier curve results. Also, while not very useful for display on a physical computer screen, by adding extra terms, a Bézier [[tetrahedron]] or Bézier [[polytope]] results.
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| Due to the nature of the equation, the entire triangle will be contained within the volume surrounded by the control points, and [[affine transformation]]s of the control points will correctly transform the whole triangle in the same way.
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| ===Halving a cubic Bézier triangle===
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| An advantage of Bézier triangles in computer graphics is, they are smooth, and can easily be approximated by regular triangles, by [[recursion|recursively]] dividing the Bézier triangle into two separate Bézier triangles, until they are considered sufficiently small, using only addition and division by two, not requiring any [[floating point]] arithmetic whatsoever.
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| The following computes the new control points for the half of the full Bézier triangle with the corner α<sup>3</sup>, a corner halfway along the Bézier curve between α<sup>3</sup> and β<sup>3</sup>, and the third corner γ<sup>3</sup>.
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| :<math>
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| \begin{pmatrix}
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| \boldsymbol{\alpha^3}'\\
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| \boldsymbol{\alpha^2\beta}'\\
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| \boldsymbol{\alpha\beta^2}'\\
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| \boldsymbol{\beta^3}'\\
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| \boldsymbol{\alpha^2\gamma}'\\
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| \boldsymbol{\alpha\beta\gamma}'\\
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| \boldsymbol{\beta^2\gamma}'\\
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| \boldsymbol{\alpha\gamma^2}'\\
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| \boldsymbol{\beta\gamma^2}'\\
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| \boldsymbol{\gamma^3}'
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| \end{pmatrix}=\begin{pmatrix}
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| 1&0&0&0&0&0&0&0&0&0\\
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| {1\over 2}&{1\over 2}&0&0&0&0&0&0&0&0\\
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| {1\over 4}&{2\over 4}&{1\over 4}&0&0&0&0&0&0&0\\
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| {1\over 8}&{3\over 8}&{3\over 8}&{1\over 8}&0&0&0&0&0&0\\
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| 0&0&0&0&1&0&0&0&0&0\\
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| 0&0&0&0&{1\over 2}&{1\over 2}&0&0&0&0\\
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| 0&0&0&0&{1\over 4}&{2\over 4}&{1\over 4}&0&0&0\\
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| 0&0&0&0&0&0&0&1&0&0\\
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| 0&0&0&0&0&0&0&{1\over 2}&{1\over 2}&0\\
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| 0&0&0&0&0&0&0&0&0&1
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| \end{pmatrix}\cdot\begin{pmatrix}
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| \boldsymbol{\alpha^3}\\
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| \boldsymbol{\alpha^2\beta}\\
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| \boldsymbol{\alpha\beta^2}\\
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| \boldsymbol{\beta^3}\\
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| \boldsymbol{\alpha^2\gamma}\\
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| \boldsymbol{\alpha\beta\gamma}\\
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| \boldsymbol{\beta^2\gamma}\\
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| \boldsymbol{\alpha\gamma^2}\\
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| \boldsymbol{\beta\gamma^2}\\
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| \boldsymbol{\gamma^3}
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| \end{pmatrix}</math>
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| :equivalently, using addition and division by two only,
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| :
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| {|
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| |-----
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| | align="center" |
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| {|
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| |-----
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| | ''β''<sup>3</sup> := (''αβ''<sup>2</sup> + ''β''<sup>3</sup>)/2
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| |-----
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| | ''αβ''<sup>2</sup> := (''α''<sup>2</sup>''β'' + ''αβ''<sup>2</sup>)/2
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| | ''β''<sup>3</sup> := (''αβ''<sup>2</sup> + ''β''<sup>3</sup>)/2
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| |-----
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| | ''α''<sup>2</sup>''β'' := (''α''<sup>3</sup> + ''α''<sup>2</sup>''β'')/2
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| | ''αβ''<sup>2</sup> := (''α''<sup>2</sup>''β'' + ''αβ''<sup>2</sup>)/2
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| | ''β''<sup>3</sup> := (''αβ''<sup>2</sup> + ''β''<sup>3</sup>)/2
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| |}
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| |-----
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| | align="center" |
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| {|
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| |-----
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| | ''β''<sup>2</sup>''γ'' := (''αβγ'' + ''β''<sup>2</sup>''γ'')/2
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| |-----
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| | ''αβγ'' := (''α''<sup>2</sup>''γ'' + ''αβγ'')/2 ||
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| | β<sup>2</sup>γ:=(αβγ+β<sup>2</sup>γ)/2
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| |}
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| |-----
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| | align="center" |
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| {|
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| |-----
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| | ''βγ''<sup>2</sup> := (''αγ''<sup>2</sup> + ''βγ''<sup>2</sup>)/2
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| |}
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| |}
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| :where := means to replace the vector on the left with the vector on the right.
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| :Note that halving a bézier triangle is similar to halving Bézier curves of all orders up to the order of the Bézier triangle.
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| ==''n''th-order Bézier triangle==
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| It is also possible to create [[quadratic equation|quadratic]] or other degrees of Bézier triangles, by changing the exponent in the original equation, in which case there will be more or fewer control points. With the exponent 1 (one), the resulting Bézier triangle is actually a regular flat [[triangle]]. In all cases, the edges of the triangle will be Bézier curves of the same degree.
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| A general ''n''th-order Bézier triangle has (''n'' + 1)(''n'' + 2)/2 control points ''a''<sup> ''i''</sup> ''β''<sup> ''j''</sup> ''γ''<sup> ''k''</sup> where ''i'', ''j'', ''k'' are nonnegative integers such that ''i'' + ''j'' + ''k'' = ''n''. The surface is then defined as
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| : <math>
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| (\alpha s + \beta t + \gamma u)^n
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| = \sum_{\begin{smallmatrix} i+j+k=n \\ i,j,k \ge 0\end{smallmatrix}} {n \choose i\ j\ k } s^i t^j u^k \alpha^i \beta^j \gamma^k
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| = \sum_{\begin{smallmatrix} i+j+k=n \\ i,j,k \ge 0\end{smallmatrix}} \frac{n!}{i!j!k!} s^i t^j u^k \alpha^i \beta^j \gamma^k
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| </math> | |
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| for all nonnegative real numbers ''s'' + ''t'' + ''u'' = 1.
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| ==See also==
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| * [[Bézier curve]]
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| * [[Bézier surface]] (biquadratic patches are Bézier rectangles)
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| * [[Surface]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| * [http://www.graphicshardware.org/previous/www_1998/presentations/bruijns/index.htm Quadratic Bézier Triangles As Drawing Primitives] Contains more info on planar and quadratic Bézier triangles.
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| * [http://othes.univie.ac.at/11497/ Paper about the use of cubic Bézier patches in raytracing (German)]
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| * {{cite web | title = Ray Tracing Triangular Bézier Patches | id = {{citeseerx|10.1.1.18.5646}} }}
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| * {{cite web | title = Triangular Bézier Clipping | id = {{citeseerx|10.1.1.62.8062}} }}
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| * [http://alex.vlachos.com/graphics/CurvedPNTriangles.pdf Curved PN triangles (a special kind of cubic Bézier triangles)]
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| * [http://www.mpi-inf.mpg.de/~mschwarz/papers/pscurvedtris-sig06.pdf Pixel-Shader-Based Curved Triangles]
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| * {{cite web | title = Surface Construction with Near Least Square Acceleration based on Vertex Normals on Triangular Meshes | id = {{citeseerx|10.1.1.6.2521}} }}
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| {{DEFAULTSORT:Bezier Triangle}}
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| [[Category:Surfaces]]
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| [[Category:Multivariate interpolation]]
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I gained a lot of fat when I was pregnant with my son, and my body happened to shop simply regarding every ounce left over for the next four years. It didn't feel wise understanding I hadn't loss anything, plus I'd had enough of feeling fat all the time. After seeing how heavy I actually looked inside my brother's wedding images, I decided to lose fat by setting a New Year's resolution!
Weight reduction is a main issue for nearly all of the folks considering it causes wellness risks. So should you join CAL system, a problem is solved easily and securely. During the course of the system, you are able to have unlimited access to the physician and health staff. You can see fat reduction clinics where the doctor will guide we at every step accurately. You can reach know which the rate of fat reduction change inside the body in an ongoing basis. Your weekly follow-up to the clinic can give we accurate information regarding your progress. The doctor might guide you proper exercises or when you want to have fast fat reduction, then the doctor might guide we thus by providing you with certain medicines.
Total body fat percentage consists of fat imperative to the metabolism in addition to stored fat. Essential fat is indispensable in terms of sustaining life and reproductive functions of the body. The imperative percentage is better for women than which for guys considering their bodies have to deal with the demands of childbearing plus additional hormonal functions. According to the accepted norms, necessary fat is 2-5% for men plus 10-13% for women. The minimal total body fat percentage suggested by healthcare experts is much more than this imperative fat percentage. Storage fat pertains to the fat accumulation in adipose tissue. Some of the adipose tissue is imperative for safeguarding internal organs in the chest and abdomen plus it is slightly high for females than for men.
The BMI index is broken down into four leading categories. Every category is represents a level of morbidity plus mortality data or wellness risk. If the bmi calculator reads lower than 18.5 you're considered underweight, if it reads between 18.5 to 25 you're at an optimal weight; if the bmi calculator reads 25 to 30 your are obese, if it reads 30 or above than we are considered overweight (over 40 morbidly obese). Patients that score above a 30 are a major concern of the healthcare profession. Studies have shown which being obese may have an impact on the total health. These same studies show how weight is shortening the average existence span.
This is the case with a person i learn. She's 63, 5'9" plus really close to 300 lbs. While she can walk, in a limited way, her efforts to lose weight are just not working. She has alternative problems also like CHF plus osteoporosis. Her target weight is around 180, thus thats over 100 lbs she must loose. I devised this program for her!
Another limitation is that age is not considered when calculating BMI. So for elderly people, even if the BMI indication is regular, the truth remains which many muscles have been lost due to age.
These formulas aren't actual, nevertheless they provide fairly exact approximations. This usually give we the ability to measure the body fat plus have a better idea of how healthy you are. Use this info perfectly, plus keep up the wise work!