@allmaps/transform

@allmaps/transform

This module serves to transform Points, LineStrings, Polygons and other spatial features from a cartesian (x, y) source plane to a destination plane. It does this using a set of Control Points, who's coordinates are known in both planes, and a specific transformation algorithm.

It is used in @allmaps/render and @allmaps/tileserver, two packages where we produce a georeferenced image by triangulating a IIIF image and drawing these triangles on a map in a specific new location, with the triangle's new vertex location computed by the transformer of this package. The transformer is constructed from Control Points in the annotation and transforms Points from the resource coordinate space of a IIIF Resource to the geo coordinate space of an interactive map.

Care was taken to make this module usable and useful outside of the Allmaps context as well! Feel free to incorporate it in your project.

How it works

This package exports the GcpTransformer class. Its instances (called 'transformers') are built from a set of Ground Control Points (GCPs) and a specified transformation type. Using these, a forward and backward transformation can be built that maps arbitrary Points in one plane to the corresponding Points in the other plane. The transformer has dedicated functions that use this transformation to transform Points and more complex geometries like LineStrings and Polygons.

Installation

This is an ESM-only module that works in browsers and in Node.js.

Install with npm:

npm install @allmaps/transform

Usage

Point

import { GcpTransformer } from '@allmaps/transform'

const transformGcps3 = [
  {
    source: [518, 991],
    destination: [4.9516614, 52.4633102]
  },
  {
    source: [4345, 2357],
    destination: [5.0480391, 52.5123762]
  },
  {
    source: [2647, 475],
    destination: [4.9702906, 52.5035815]
  }
]

const transformer = new GcpTransformer(transformGcps3, 'helmert')

const transformedPoint = transformer.transformForward([100, 100])
// transformedPoint = [4.9385700843392435, 52.46580484503631]

const roundtripTransformedPoint = transformer.transformBackward([
  4.9385700843392435, 52.46580484503631
])
// roundtripTransformedPoint = [100, 100]

LineString

In this example we transform backward, and from a GeoJSON Geometry.

const transformGcps7 = [
  {
    source: [0, 0],
    destination: [0, 0]
  },
  {
    source: [100, 0],
    destination: [20, 0]
  },
  {
    source: [200, 100],
    destination: [40, 20]
  },
  {
    source: [200, 200],
    destination: [40, 40]
  },
  {
    source: [150, 250],
    destination: [40, 100]
  },
  {
    source: [100, 200],
    destination: [20, 40]
  },
  {
    source: [0, 100],
    destination: [0, 20]
  }
]

const options = {
  maxOffsetRatio: 0.001,
  maxDepth: 2
}
// We transform backward (from destination to source) and have GeoJSON input.
// Hence `destinationIsGeographic: true` will be set automatically

const transformer = new GcpTransformer(transformGcps7, 'polynomial')

const lineStringGeoJSON = {
  type: 'LineString',
  coordinates: [
    [10, 50],
    [50, 50]
  ]
}

const transformedLineString = transformer.transformBackward(
  lineStringGeoJSON,
  options
)
// transformedLineString = [
//   [31.06060606060611, 155.30303030303048],
//   [80.91200458875993, 165.7903106766409],
//   [133.1658635549907, 174.5511756850417],
//   [185.89024742146262, 181.22828756380306],
//   [237.12121212121218, 185.60606060606085]
// ]

// Notice how the result has two layers of midpoints!
// In a first step the Point [133.16, 174.55] is added between the start and end Point
// Then [80.91, 165.79] and [185.89, 181.22] are added in between.

Polygon

In this example we transform to a GeoJSON Geometry.

const transformGcps6 = [
  {
    source: [1344, 4098],
    destination: [4.4091165, 51.9017125]
  },
  {
    source: [4440, 3441],
    destination: [4.5029222, 51.9164451]
  },
  {
    source: [3549, 4403],
    destination: [4.4764224, 51.897309]
  },
  {
    source: [1794, 2130],
    destination: [4.4199066, 51.9391509]
  },
  {
    source: [3656, 2558],
    destination: [4.4775683, 51.9324358]
  },
  {
    source: [2656, 3558],
    destination: [4.4572643, 51.9143043]
  }
]

const options = {
  maxOffsetRatio: 0.00001,
  maxDepth: 1
}

const transformer = new GcpTransformer(transformGcps6, 'thinPlateSpline')

const polygon = [
  [
    [1000, 1000],
    [1000, 2000],
    [2000, 2000],
    [2000, 1000]
  ]
]

const transformedPolygonGeoJSON = transformer.transformForwardAsGeojson(
  polygon,
  options
)
// const transformedPolygonGeoJSON = {
//   type: 'Polygon',
//   coordinates: [
//     [
//       [4.388957777030093, 51.959084191571606],
//       [4.390889520773774, 51.94984430356657],
//       [4.392938913951547, 51.94062947962427],
//       [4.409493277493718, 51.94119110133424],
//       [4.425874493300959, 51.94172557475595],
//       [4.4230497784967655, 51.950815146974556],
//       [4.420666790347598, 51.959985351835975],
//       [4.404906205946158, 51.959549039424715],
//       [4.388957777030093, 51.959084191571606]
//     ]
//   ]
// }

MultiPoint

In this example we transform a MultiPoint to a MultiPoint.

const transformGcps7 = [
  {
    source: [0, 0],
    destination: [0, 0]
  },
  {
    source: [100, 0],
    destination: [20, 0]
  },
  {
    source: [200, 100],
    destination: [40, 20]
  },
  {
    source: [200, 200],
    destination: [40, 40]
  },
  {
    source: [150, 250],
    destination: [40, 100]
  },
  {
    source: [100, 200],
    destination: [20, 40]
  },
  {
    source: [0, 100],
    destination: [0, 20]
  }
]

const options = {
  inputIsMultiGeometry: true // this assures the transform method recognises the input as a multiPoint, not a LineString
}

const transformer = new GcpTransformer(transformGcps7, 'polynomial')

const multiPoint = [
  [10, 50],
  [50, 50]
]

const transformedMultiPoint = transformer.transformForward(multiPoint, options)
// const transformedMultiPoint = [
//   [31.06060606060611, 155.30303030303048],
//   [237.12121212121218, 185.60606060606085]
// ]

Transformation types

A transformer is build from a set of GCPs and a transformation type. The following transformation types are supported.

	Type	Description	Properties	Minimum number of GCPs
	straight	Straight transformation	Applies translation and scaling. Preserves shapes and angles.	2
	helmert	Helmert transformation or 'similarity transformation'	Applies translation, scaling and rotation. Preserves shapes and angles.	2
	polynomial (default), also polynomial1	First order polynomial transformation	Applies translation, scaling, rotation and shearing. Preserves lines and parallelism.	3
	polynomial2	Second order polynomial transformation.	Applies second order effects. Adds some bending flexibility.	6
	polynomial3	Third order polynomial transformation	Applies third order effects. Adds more bending flexibility.	10
	thinPlateSpline	Thin Plate Spline transformation or 'rubber sheeting' (with affine part)	Applies smooth transformation. Transformation is 'exact' at GPCs. (see this notebook)	3
	projective	Projective or 'perspective' transformation, used for aerial images	Follow perspective rules. Preserves lines and cross-ratios.	4

Transformer methods

Once a transformer is built, it can be used to transform geometries forward and backward.

All transformer methods accepts Points, LineStrings as well as Polygons (and MultiPoints, MultiLineStrings and MultiPolygons), both as standard geometries or GeoJSON geometries. There are, however, separate methods for transforming to standard geometries or to GeoJSON geometries. There are also separate methods for transforming forward or backward.

Hence, the main methods are: transformForward(), transformForwardAsGeojson(), transformBackward() and transformBackwardAsGeojson()

Alternatively the same four methods are available with more expressive term for the Allmaps use case: replacing Forward by ToGeo and Backward by ToResource. E.g.: transformToGeoAsGeojson().

Transform options

Some options are available to improve transformations, e.g. to transform LineStrings or Polygons by recursively adding midpoints, or to correctly deal with a possible different handedness of source and destination coordinates.

These options can be specified when using a transformer's method to transform geometries, or earlier upon the creation of the transformer. Options specified in a transformer's method override options specified during the transformer's creation, which in term override the options derived from the data format (e.g. setting 'true' when source is GeoJSON), which in term override the default options.

The differentHandedness option is used both when a transformer and when a geometry is transformed, and should not be altered between these two actions.

Here's an overview of the available options:

Option	Description	Type	Default
maxOffsetRatio	Maximum offset ratio when recursively adding midpoints (smaller means more midpoints)	number	0
minOffsetDistance	Minimum offset distance when recursively adding midpoints (higher means more midpoints)	number	Infinity
minLineDistance	Minimum line distance when recursively adding midpoints (higher means more midpoints)	number	Infinity
maxDepth	Maximum recursion depth when recursively adding midpoints (higher means more midpoints)	number	0 (i.e. no midpoints by default!)
sourceIsGeographic	Use geographic distances and midpoints for lon-lat source points	boolean	false (true when source is GeoJSON)
destinationIsGeographic	Use geographic distances and midpoints for lon-lat destination points	boolean	false (true when destination is GeoJSON)
inputIsMultiGeometry	Whether the input should be considered as a MultiPoint, MultiLineString or MultiPolygon. This is necessary since the standard geometry (as opposed to GeoJSON geometries) types are not deterministic: the types of LineString and MultiPoint are identical.	boolean	false
differentHandedness	Whether one of the axes should be flipped while computing the transformation parameters. Should be true if the handedness differs between the source and destination.	boolean	false
evaluationType	Whether to evaluate the transformation function or one of it's derivatives.	'function' | 'partialDerivativeX' | 'partialDerivativeY'	'function'

Recursively adding midpoints

When transforming LineStrings and Polygons, it can happen that simply transforming every Point is not sufficient.

Two factors are at play which may require a more granular transformation: the transformation (which can be non-shape preserving, as is the case with all transformation in this package except for Helmert and 1st degree polynomial) or the geographic nature of the coordinates (where lines are generally meant as 'great arcs' but could be interpreted as lon-lat cartesian lines).

An algorithm will therefore recursively add midpoints in each segment (i.e. between two Points) to make the line more granular. A midpoint is added at the transformed middle Point of the original segment if all of the following conditions are met:

• That the ratio of (the distance between the middle Point of the transformed segment and the transformed middle Point of the original segment) to the length of the transformed segment, is larger than the specified maxOffsetRatio.
• The distance between the middle Point of the transformed segment and the transformed middle Point of the original segment is smaller than the specified minOffsetDistance.
• The transformed segment is smaller than the specified minLineDistance.
• The process hasn't repeated more than maxDepth iterations.

The computation of the midpoints and distances in the source and destination domains during this process uses geometric algorithms, unless sourceIsGeographic or destinationIsGeographic are set to true, in which case geographic algorithms (such as 'Great-circle distance') are used.

Handedness

For some transformations, it is important that the source and destination planes have the same handedness.

When we consider 2D Cartesian planes, there are two types of 'handedness'. A Cartesian plane with the positive x-axis pointing right and the positive y-axis pointing up (and the x-axis being the "first" and the y-axis the "second" axis) is said to have right-handed orientation (also called standard, positive or counter-clockwise). This is for example the case in the equirectangular projection - at least if the coordinate order is (lon, lat). Alternatively, if the y-axis points downwards, we say the orientation is left-handed (or negative or clock-wise). This is for example the case for typical pixel coordinates, which have their origin in the top left corner.

The handedness of the source and destination can differ, for example if the source are pixels of an image and the destination are (lon, lat) coordinates (which is the typical case for Allmaps). For many transformations a separate transformation is computed for both axes and hence it does not matter whether the source and destination have the same handedness. For some transformations, like the Helmert transformation, the transformation of X and Y coordinates are computed jointly (they are said to be 'coupled') and the difference matters. The algorithms won't produce the desired results unless action is taken to align the handedness.

Therefore, in case the handedness differs one can set the differentHandedness parameter to true. This will internally flip the y-axis of the source so as to align the handedness of both during computation.

Distortions

Some transformations may induce distortions. Let's consider transforming an image to make this more visual. It we take a Helmert transformation of an image, we will see that it doesn't distort the image much: it will scale, rotate and translate the image, but not shear it (angles are preserved) - the only distortion applied is the scaling, and that scaling is the same everywhere across the image. If, on the other hand, we take a Thin Plate Spline transformation (with many GCPs) of that same image, we will see that the image will be distorted much, and will look like a rubber sheet which has been pulled and deformed in many different locations. Every pixel will be distorted in a unique way, such that both the areas and angles of the original image are not preserved.

We can compute these distortions locally, at every point. The approach implemented here is based on the theory of 'Differential Distortion Analysis': by evaluating the partial derivatives of the transformation function at every point we can compute local distortion measures from these derivatives, such as the area distortion log2sigma and angular distortion twoOmega. These will tell us how much the area and angles are distortion at every point. Thereafter averaging over all points can give un an indication of the overall distortion.

'Differential Distortion Analysis' was earlier implemented in this Matlab/Octave package following peer reviewed publications of both the theoretical approach an an application to a historical map.

This packages supports the evaluation of the partial derivatives in the transformForward() and transformBackward() functions via their transform options, and exports a function computeDistortionFromPartialDerivatives() to compute the distortion measures from these partial derivatives. The supported distortion measures are available via the exported supportedDistortionMeasures constant. These include:

Key	Type	Description	Example
log2sigma	Area distortion measure	The base-2 logarithm of the area scale factor σ, which indicates how much a local infinitesimal surface element is enlarged on the map (relative to the map’s scale).	0 for no area distortion, 1 if the area is twice as big, -1 if the are is twice as small after transformation.
twoOmega	Angular distortion measure	The maximum angular distortion 2Ω, which indicated the maximal (taken over all possible angles between two direction from that point) difference between an angle before and after the transformation, making it a measure for shearing.	0 for no angular distortion, >0 for angular distortion.
airyKavr	Airy-Kavrayskiy distortion measure	A measure combining the effects of areal and angular distortion.	0 for no distortion, >0 for distortion.
signDetJ	Flip measure	The transformation's Jacobian determinant flipping sign, describing 'fold-over' of the transformation.	1 for no flip, -1 for flip.
thetaa	Tissot indicatrix axis	The angle between the major axis of the Tissot indicatrix and the cartesian x-axis.	0 for no rotation, >0 for rotation.

Here's an example on how to compute local distortion.

import { GcpTransformer, computeDistortionFromPartialDerivatives } from '@allmaps/transform'

const transformGcps6 = ... // See above

const helmertTransformer = new GcpTransformer(transformGcps6, 'helmert')
helmertTransformer.createForwardTransformation()
const referenceScale = helmertTransformer.forwardTransformation.scale

const transformer = new GcpTransformer(transformGcps6, 'thinPlateSpline')
const input = [1000, 1000]
const partialDerivativeX = transformer.transformForward(input, {
  evaluationType: 'partialDerivativeX'
})
const partialDerivativeY = transformer.transformForward(input, {
  evaluationType: 'partialDerivativeY'
})
const distortion = computeDistortionFromPartialDerivatives(
  partialDerivativeX,
  partialDerivativeY,
  'log2sigma',
  referenceScale
)
// distortion = 1.7800137112938559
// => At this input location the area has significantly expanded after the transformation

Notes

Typing

GCPs

GCPs can be supplied as an array of objects containing source and destination coordinates:

type TransformGcp = {
  source: [number, number]
  destination: [number, number]
}

Or you can supply an array of objects containing resource and geo coordinates. This is the format used in Georeference Annotations:

type Gcp = {
  resource: [number, number]
  geo: [number, number]
}

Geometry types

Standard geometries: the following geometry types are used by default in this and other packages.

type Point = [number, number]

type LineString = Point[]

type Polygon = Point[][]
// A Polygon is an array of rings of at least three points
// Rings are not closed: the first point is not repeated at the end.
// There is no requirement on winding order.

type MultiPoint = Point[]
// Notice that this is equivalent to the LineString type, hence the `inputIsMultiGeometry` option

type MultiLineString = Point[][]
// Notice that this is equivalent to the Polygon type, hence the `inputIsMultiGeometry` option

type MultiPolygon = Point[][][]

type Geometry =
  | Point
  | LineString
  | Polygon
  | MultiPoint
  | MultiLineString
  | MultiPolygon

GeoJSON geometries follow the GeoJSON specification.

SVG geometries are expressed using the following types (but note that some functions allow svg's to be passed as a string):

export type SvgCircle = {
  type: 'circle'
  attributes?: SvgAttributes
  coordinates: Point
}

export type SvgLine = {
  type: 'line'
  attributes?: SvgAttributes
  coordinates: [Point, Point]
}

export type SvgPolyLine = {
  type: 'polyline'
  attributes?: SvgAttributes
  coordinates: Point[]
}

export type SvgPolygon = {
  type: 'polygon'
  attributes?: SvgAttributes
  coordinates: Point[]
}

export type SvgRect = {
  type: 'rect'
  attributes?: SvgAttributes
  coordinates: Point[]
}

export type SvgGeometry =
  | SvgCircle
  | SvgLine
  | SvgPolyLine
  | SvgPolygon
  | SvgRect

Transform vs. GDAL

The transformation algorithms of this package correspond to those of GDAL and the results are (nearly) identical. See the tests for details.

For a little history: this library started out as a JavaScript port of gdaltransform (as described in this notebook) and initially only implemented polynomial transformations of order 1. Later Thin Plate Spline transformations were added (see this notebook) amongst other transformations, which lead to a refactoring using the ml-matrix library. This library is used for creating and solving the linear systems of equations that are at the heart of each of each of these transformations.

Notes

• Only linearly independent control points should be considered when checking if the criterion for the minimum number of control points is met. For example, three control points that are collinear (one the same line) only count as two linearly independent points. The current implementation doesn't check such linear (in)dependance, but building a transformer with insufficient linearly independent control points will result in a badly conditioned matrix (no error but diverging results) or non-invertible matrix (error when inverting matrix).
• The transform functions are map-projection agnostic: they describe a transformation for one cartesian (x, y) plane to another. Using control points with (longitude, latitude) coordinates will produce a transformation from or to the cartesian plane of an equirectangular projection. (The only semi-exception to this is when using the destinationIsGeographic and sourceIsGeographic parameters - although these consider coordinates as lying on a sphere more than as projection coordinates.)

CLI

The @allmaps/cli package creates and interface for four specific use cases:

• Transforming points to points.
• Transforming SVG geometries from the resource coordinates space of a IIIF resource to GeoJSON objects in the geo coordinate space of an interactive map.
• Transforming GeoJSON objects from the geo coordinate space of an interactive map to SVG objects in the resource coordinates space of a IIIF resource, given (the GCPs and transformation type from) a Georeference Annotation
• Vice versa: transforming SVG objects from the resource coordinates to GeoJSON objects in the geo coordinate space.
• Transforming the SVG resource mask included in a Georeference Annotation to a GeoJSON Polygon.

Benchmark

Here are some benchmarks on building and using a transformer, as computed on a 2023 MacBook Air M2.

Creating a transformer (with 10 points) (and transform 1 point)

Type	Options	Ops/s
helmert		63499
polynomial	order: 1	133824
polynomial	order: 2	66501
polynomial	order: 3	26750
thinPlateSpline		20487
projective		27581

Using a transformer (with 10 points) to transform 1 point

Type	Options	Ops/s
helmert		21612153
polynomial	order: 1	19993234
polynomial	order: 2	19887376
polynomial	order: 3	3930665
thinPlateSpline		2931361
projective		20386139

See ./bench/index.js.

The benchmark can be run with pnpm run bench.

API

Table of Contents

• allmaps/transform
• GcpTransformer
  • Parameters
  • createForwardTransformation
  • createBackwardTransformation
  • transformForward
  • transformForwardAsGeojson
  • transformBackward
  • transformBackwardAsGeojson
  • transformToGeo
  • transformToGeoAsGeojson
  • transformToResource
  • transformToResourceAsGeojson
  • transformSvgToGeojson
  • transformSvgStringToGeojsonFeatureCollection
  • transformGeojsonToSvg
  • transformGeojsonFeatureCollectionToSvgString
• Transformation
  • Parameters
• computeDistortionFromPartialDerivatives
  • Parameters

allmaps/transform

GcpTransformer

A Ground Control Point Transformer, containing a forward and backward transformation and specifying functions to transform geometries using these transformations.

Parameters

• gcps (Array<TransformGcp> | Array<Gcp>) An array of Ground Control Points (GCPs)
• type TransformationType The transformation type (optional, default 'polynomial')
• options

createForwardTransformation

Create forward transformation

createBackwardTransformation

Create backward transformation

transformForward

Transforms a Geometry or a GeoJSON geometry forward to a Geometry

Parameters

• input (Geometry | GeojsonGeometry) Geometry or GeoJSON geometry to transform
• options Partial<TransformOptions>? Transform options

Returns Geometry Forward transform of input as Geometry

transformForwardAsGeojson

Transforms a Geometry or a GeoJSON geometry forward to a GeoJSON geometry

Parameters

• input (Geometry | GeojsonGeometry) Geometry or GeoJSON geometry to transform
• options Partial<TransformOptions>? Transform options

Returns GeojsonGeometry Forward transform of input, as GeoJSON geometry

transformBackward

Transforms a geometry or a GeoJSON geometry backward to a Geometry

Parameters

• input (Geometry | GeojsonGeometry) Geometry or GeoJSON geometry to transform
• options Partial<TransformOptions>? Transform options

Returns Geometry backward transform of input, as geometry

transformBackwardAsGeojson

Transforms a Geometry or a GeoJSON geometry backward to a GeoJSON geometry

Parameters

• input (Geometry | GeojsonGeometry) Geometry or GeoJSON geometry to transform
• options Partial<TransformOptions>? Transform options

Returns GeojsonGeometry backward transform of input, as GeoJSON geometry

transformToGeo

Transforms Geometry or GeoJSON geometry forward, as Geometry

Parameters

• input (Geometry | GeojsonGeometry) Input to transform
• options Partial<TransformOptions>? Transform options

Returns Geometry Forward transform of input, as Geometry

transformToGeoAsGeojson

Transforms a Geometry or a GeoJSON geometry forward, to a GeoJSON geometry

Parameters

• input (Geometry | GeojsonGeometry) Input to transform
• options Partial<TransformOptions>? Transform options

Returns Geometry Forward transform of input, as GeoJSON geometry

transformToResource

Transforms a Geometry or a GeoJSON geometry backward, to a Geometry

Parameters

• input (Geometry | GeojsonGeometry) Input to transform
• options Partial<TransformOptions>? Transform options

Returns Geometry Backward transform of input, as a Geometry

transformToResourceAsGeojson

Transforms a Geometry or a GeoJSON geometry backward, to a GeoJSON geometry

Parameters

• input (Geometry | GeojsonGeometry) Input to transform
• options Partial<TransformOptions>? Transform options

Returns GeojsonGeometry Backward transform of input, as a GeoJSON geometry

transformSvgToGeojson

Transforms a SVG geometry forward to a GeoJSON geometry

Note: Multi-geometries are not supported

Parameters

• geometry SvgGeometry SVG geometry to transform
• options Partial<TransformOptions>? Transform options

Returns GeojsonGeometry Forward transform of input, as a GeoJSON geometry

transformSvgStringToGeojsonFeatureCollection

Transforms a SVG string forward to a GeoJSON FeatureCollection

Note: Multi-geometries are not supported

Parameters

• svg string SVG string to transform
• options Partial<TransformOptions>? Transform options

Returns GeojsonFeatureCollection Forward transform of input, as a GeoJSON FeatureCollection

transformGeojsonToSvg

Transforms a GeoJSON geometry backward to a SVG geometry

Note: Multi-geometries are not supported

Parameters

• geometry GeojsonGeometry GeoJSON geometry to transform
• options Partial<TransformOptions>? Transform options

Returns SvgGeometry Backward transform of input, as SVG geometry

transformGeojsonFeatureCollectionToSvgString

Transforms a GeoJSON FeatureCollection backward to a SVG string

Note: Multi-geometries are not supported

Parameters

• geojson GeojsonFeatureCollection GeoJSON FeatureCollection to transform
• options Partial<TransformOptions>? Transform options

Returns string Backward transform of input, as SVG string

Transformation

Transformation class. Abstract class, extended by the various transformations.

Parameters

• sourcePoints Array<Point> The source points
• destinationPoints Array<Point> The destination points
• type TransformationType The transformation type
• pointCountMinimum number The minimum number of points for the transformation type

computeDistortionFromPartialDerivatives

Compute distortion from partial derivatives

Parameters

• partialDerivativeX Point the partial derivative to 'x' of the transformation, evaluated at a set point
• partialDerivativeY Point the partial derivative to 'x' of the transformation, evaluated at a set point
• distortionMeasure DistortionMeasure? the requested distortion measure, or undefined to return 0
• referenceScale number? the reference area scaling (sigma) to take into account, e.g. computed via a helmert transform (optional, default 1)

Returns number the distortion measure at the set point