libE57 Coordinate systems

This is a list of the coordinate systems used in the E2807 standard and the libE57 library.

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• XYZ - Cartesian Coordinates

X, Y, Z - Points in cartesian coordinates are represented by an ordered triple (x, y, z) where x, y and z are coordinates along the X, Y, Z axis, respectively. The Z-axis is in the UP direction, the Y-axis is the northing and X-axis is the easting. The coordinate system is right-handed and is measured in meters. If a lidar scanner uses a left-handed system, the data should be converted into this right-handed definition.



X,Y,Z Cartesian Coordinates

• CYL - Cylindrical Coordinates

Radial (ρ), Azimuth (θ), Z - Points in cylindrical coordinates are represented by an ordered triplet (ρ,θ,z), where ρ is the radial distance (in meters), θ is the azimuth angle (in radians), and z is the height (in meters).

The following restrictions on cylindrical coordinates are applied:

ρ >= 0

-π < θ <= π



Radial, Azimuth, and Z Coordinates

• RAE - Spherical Coordinates

Range (r), Azimuth (θ), Elevation (φ) - Points in spherical coordinates are represented by an ordered triplet ( r, θ, φ), where r is the range (in meters), θ is the azimuth angle (in radians), and φ is the elevation angle (in radians). The azimuth angle is measured as the counterclock-wise rotation of the positive X-axis about the positive Z-axis of a Cartesian reference frame.

The following restrictions on spherical coordinates are applied:

r >= 0

-π < θ <= π

-π/2 <= φ <= π/2



Range, Azimuth, and Elevation Coordinates

• Conversion between XYZ and CYL Coordinates

The conversion between cylindrical and cartesian coordinates are accomplished through the formulas:

x = ρ cos( θ )

y = ρ sin( θ )

z = z

ρ = √ x² + y²

θ = arctan2( y, x )

z = z

• Conversion between XYZ and RAE Coordinates

The conversion between spherical and cartesian coordinates are accomplished through the formulas:

x = r cos( φ ) cos( θ )

y = r cos( φ ) sin( θ )

z = r sin( φ )

r = √ x² + y² + z²

θ = arctan2( y, x )

φ = arcsin( z / r )

• Scanner Coordinate System

The Scanner's coordinate system is defined by the manufacturer of the scanner. The Pose/Translation vector is the position of the scanner's nodal point (origin) in the World Coordinate System of the E57 file.



Scanner Coordinate System

• Pinhole Camera Projection Model

Pinhole representation stores an image that is mapped from 3D using a pinhole projection model. Digital cameras with typical, non-fisheye, lenses are well-approximated by this model.

Given a point (x, y, z) in cartesian coordinates in the camera frame of reference, where z < 0, the image coordinates (Ximage, Yimage) are given by the following equations of projection:

Ximage = principalPointX - ( x / z) (focalLength / pixelWidth)

Yimage = principalPointY - ( y / z) (focalLength / pixelHeight)



Pinhole Camera Projection Model

The model allows 3D points to be projected onto camera images and image pixels to be projected into 3D space.

• Spherical Camera Projection Model

Spherical representation stores an image that is mapped from 3D using a spherical projection model. Image from fisheye lenses or image mosaics generated from a single position can be represented using this model.

Given a point in spherical coordinates (r, θ, φ), the image coordinates (Ximage, Yimage) are given by the following equations of projection:

Ximage = imageWidth / 2 - θ / pixelWidth

Yimage = imageHeight / 2 - φ / pixelHeight

Image coordinate (0,0) is the top, left corner of the pixel at the top, left corner of the image.



Spherical Camera Projection Model

• Cylindrical Camera Projection Model

Cylindrical representation stores an image that is mapped from 3D using a cylindrical projection model. Images from a rotating single line scanning camera can be represented using this model.

Given a point in cylindrical coordinates (ρ, θ, z), the image coordinates (Ximage, Yimage) are given by the following equations of projection:

Ximage = imageWidth / 2 - θ / pixelWidth

Yimage = principalPointY - z (radius / pixelHeight) / ρ



Cylindrical Camera Projection Model

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