Euler

API for initializing and manipulating euler angles.

Euler angles are a simple representation of a 3 dimensional rotation; comprised of 3 ordered heading, pitch and roll rotations. An important thing to understand is that the axis of rotation belong to the object being rotated and so they also rotate as each of the heading, pitch and roll rotations are applied.

One way to consider euler angles is to imagine controlling an aeroplane, where you first choose a heading (Such as flying south east), then you set the pitch (such as 30 degrees to take off) and then you might set a roll, by dipping the left, wing as you prepare to turn.

They have some advantages and limitations that it helps to be aware of:

Advantages:

1. Easy to understand and use, compared to quaternions and matrices, so may be a good choice for a user interface.

2. Efficient storage, needing only 3 components any rotation can be represented.

Disadvantages:

1. Aliasing: it’s possible to represent some rotations with multiple different heading, pitch and roll rotations.

2. They can suffer from a problem called Gimbal Lock. A good explanation of this can be seen on wikipedia here: http://en.wikipedia.org/wiki/Gimbal_lock but basically two of the axis of rotation may become aligned and so you loose a degree of freedom. For example a pitch of +-90° would mean that heading and bank rotate around the same axis.

3. If you use euler angles to orient something in 3D space and try to transition between orientations by interpolating the component angles you probably wont get the transitions you expect as they may not follow the shortest path between the two orientations.

4. There’s no standard to what order the component axis rotations are applied. The most common convention seems to be what we do in clib with heading (y-axis), pitch (x-axis) and then roll (z-axis), but other software might apply x-axis, y-axis then z-axis or any other order so you need to consider this if you are accepting euler rotations from some other software. Other software may also use slightly different aeronautical terms, such as “yaw” instead of “heading” or “bank” instead of “roll”.

To minimize the aliasing issue we may refer to “Canonical Euler” angles where heading and roll are restricted to +- 180° and pitch is restricted to +- 90°. If pitch is +- 90° bank is set to 0°.

Quaternions don’t suffer from Gimbal Lock and they can be nicely interpolated between, their disadvantage is that they don’t have an intuitive representation.

A common practice is to accept angles in the intuitive Euler form and convert them to quaternions internally to avoid Gimbal Lock and handle interpolations. See c_quaternion_init_from_euler().

c_euler_t
heading

Angle to rotate around an object’s y axis

pitch

Angle to rotate around an object’s x axis

roll

Angle to rotate around an object’s z axis

Represents an ordered rotation first of @heading degrees around an object’s y axis, then @pitch degrees around an object’s x axis and finally @roll degrees around an object’s z axis.

Note

It’s important to understand the that axis are associated with the object being rotated, so the axis also rotate in sequence with the rotations being applied.

The members of a c_euler_t can be initialized, for example, with c_euler_init() and c_euler_init_from_quaternion ().

You may also want to look at c_quaternion_init_from_euler() if you want to do interpolation between 3d rotations.

void c_euler_init(c_euler_t *euler, float heading, float pitch, float roll)
Parameters:
  • euler – The c_euler_t angle to initialize
  • heading – Angle to rotate around an object’s y axis
  • pitch – Angle to rotate around an object’s x axis
  • roll – Angle to rotate around an object’s z axis

Initializes euler to represent a rotation of x_angle degrees around the x axis, then y_angle degrees around the y_axis and z_angle degrees around the z axis.

void c_euler_init_from_matrix(c_euler_t *euler, const c_matrix_t *matrix)
Parameters:
  • euler – The c_euler_t angle to initialize
  • matrix – A c_matrix_t containing a rotation, but no scaling, mirroring or skewing.

Extracts a euler rotation from the given matrix and initializses euler with the component x, y and z rotation angles.

void c_euler_init_from_quaternion(c_euler_t *euler, const c_quaternion_t *quaternion)
Parameters:
  • euler – The c_euler_t angle to initialize
  • quaternion – A c_euler_t with the rotation to initialize with

Initializes a euler rotation with the equivalent rotation represented by the given quaternion.

_Bool c_euler_equal(const void *v1, const void *v2)
Parameters:
  • v1 – The first euler angle to compare
  • v2 – The second euler angle to compare

Compares the two given euler angles v1 and v1 and it they are equal returns true else false.

Note

This function only checks that all three components rotations are numerically equal, it does not consider that some rotations can be represented with different component rotations

Returns:true if v1 and v2 are equal else false.
c_euler_t *c_euler_copy(const c_euler_t *src)
Parameters:

Allocates a new c_euler_t and initilizes it with the component angles of src. The newly allocated euler should be freed using c_euler_free().

Returns:A newly allocated c_euler_t
void c_euler_free(c_euler_t *euler)
Parameters:
  • euler – A c_euler_t allocated via c_euler_copy()

Frees a c_euler_t that was previously allocated using c_euler_copy().

void c_euler_init(c_euler_t *euler, float heading, float pitch, float roll)
Parameters:
  • euler – The c_euler_t angle to initialize
  • heading – Angle to rotate around an object’s y axis
  • pitch – Angle to rotate around an object’s x axis
  • roll – Angle to rotate around an object’s z axis

Initializes euler to represent a rotation of x_angle degrees around the x axis, then y_angle degrees around the y_axis and z_angle degrees around the z axis.

void c_euler_init_from_matrix(c_euler_t *euler, const c_matrix_t *matrix)
Parameters:
  • euler – The c_euler_t angle to initialize
  • matrix – A c_matrix_t containing a rotation, but no scaling, mirroring or skewing.

Extracts a euler rotation from the given matrix and initializses euler with the component x, y and z rotation angles.

void c_euler_init_from_quaternion(c_euler_t *euler, const c_quaternion_t *quaternion)
Parameters:
  • euler – The c_euler_t angle to initialize
  • quaternion – A c_euler_t with the rotation to initialize with

Initializes a euler rotation with the equivalent rotation represented by the given quaternion.

_Bool c_euler_equal(const void *v1, const void *v2)
Parameters:
  • v1 – The first euler angle to compare
  • v2 – The second euler angle to compare

Compares the two given euler angles v1 and v1 and it they are equal returns true else false.

Note

This function only checks that all three components rotations are numerically equal, it does not consider that some rotations can be represented with different component rotations

Returns:true if v1 and v2 are equal else false.
c_euler_t *c_euler_copy(const c_euler_t *src)
Parameters:

Allocates a new c_euler_t and initilizes it with the component angles of src. The newly allocated euler should be freed using c_euler_free().

Returns:A newly allocated c_euler_t
void c_euler_free(c_euler_t *euler)
Parameters:
  • euler – A c_euler_t allocated via c_euler_copy()

Frees a c_euler_t that was previously allocated using c_euler_copy().