Maths - Standards

I am very keen to have consistent standards of notation and terminology across this site. Its hard enough to learn these things without being confused by inconsistent standards. It may not be perfect yet so, if you find any inconsistencies across the site please let me know.

I have documented the maths notation on this page.

I have also tried to apply standards to the physics notation and terminology (see physics standards).

In some cases there are different ways to notate, calculate and display. Wherever possible I have tried to base these on standards, here are some standards used:

  This site VRML/X3D (ISO/IEC 19775) OpenGL DirectX


Vector shown as row or column       see D3DVECTOR structure  
Matrix order     opposite of everything else see D3DMATRIX structure  
Coordinate System right-handed cartesian right-handed cartesian right-handed cartesian (for orthographic projection) left-handed cartesian  
x,y and z definitions

+X to the right
+Y straight up
+Z axis toward viewer

+X to the right
+Y straight up
+Z axis toward viewer

Orthographic Projection see glOrtho()
+X to the right
+Y straight up
+Z axis toward viewer

+X = to the right
+Y = top of screen
+Z = depth buffer = distance from viewer (range 0 to 1)
W = scaling value for perspective transform

+X to the right
+Y straight up
+Z axis into screen
Euler angles Heading = rotation about y axis
Attitude = rotation about z axis
Bank = rotation about x axis
Euler angle order Heading applied first
Attitude applied second
Bank applied last
Positive angle right hand screw right hand screw right hand screw (see glRotated)    
Angle units radians radians degrees (see glRotated) radians  
Vector Cross product

right hand:

x = Ay * Bz - By * Az
y = Az * Bx - Bz * Ax
z = Ax * By - Bx * Ay

so, mechanics equations are,

v vector = w vector×r vector
torque  vector = r vector×force vector


Clifford Algebra: factor ordering


Clifford Algebra: element ordering


I have tried to use established open standards rather than inventing my own wherever I can.

Where these standards conflict I have used VRML/X3D (ISO/IEC 19775) on this site. For Euler angles I have used NASA standard Aeroplane but I have swapped the y and z axis, to conform to x3d, so:

aeroplane rotation Heading applied first (rotation about y axis – which is up-down)
Attitude applied second (rotation about z axis which is along wing)
Bank applied last (rotation about x axis which is front-back)

Although z and y are swapped: Heading, Attitude and Bank are applied in the same order so the properties of NASA standard Aeroplane are still the same, i.e. the singularity still happens at attitude= -+90 degrees.

I mentioned NASA standard Aeroplane (as its American I should have written NASA standard Airplane) . The problem is that I can't actually find this standard even from NASA employees that I have communicated with. The only information that I have about this standard is from a book: Analytical Dynamics (McGraw-Hill International Editions) and this is very limited information. Is unfortunate as a standard from such a source as NASA would be helpful.

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Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

cover This book introduces 3D concepts, VRML, Java3D, MPEG4/BIFS, and X3D. It is a very good introduction to the theory, The writers have an in depth knowledge due to their involvement in the standards making. This is a good book to help you choose which 3D open standards to use and to give you a good insight into these standards. It is probably not for those who want a basic introduction to 3D or for these who want a step-by-step programming in 3D book.

This site may have errors. Don't use for critical systems.

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