Setup

• Getting Started
• Sensor Definitions
• Bluetooth Tracking with MTR-1
• General Tips
• How To Change The Update Frequency Of Sensors/Force Update
• Zone Configuration Using Home Assistant
• Zone Configuration Using HLKRadarTool App

Getting Started

 

 

Please refer to the general getting started article

Sensor Definitions

Component

The ld2450 component enables the use of the HLK-LD2450 human presence and motion detection radar module with ESPHome.

HLK-LD2450 Official Datasheet and user manuals

Motion target tracking involves real-time tracking of the position of a moving target within a specific area, enabling measurements of distance, angle, and speed. The LD2450 is a sensor module for tracking motion targets within the Hi-link 24G millimeter-wave radar series. It features simplified 24 GHz radar sensor hardware and intelligent algorithm firmware. It is mainly used in general indoor scenarios such as homes, offices, and hotels for tracking the location of moving human targets.

HLKRadarTool Android App

HLKRadarTool iOS App

Note: Ensure that the LD2450 firmware version is V2.02.23090617 or later for proper integration functionality. You can use the HLKRadarTool mobile app to update the firmware.

The UART needs to be set up in your configuration for this sensor to work. parity and stop_bits must be NONE and 1, respectively. Using hardware UART pins is highly recommended, as they support the out-of-the-box 256000 baud rate of the LD2450 sensor.

# uart configuration
uart:
  id: uart_ld2450
  tx_pin: 17
  rx_pin: 21
  baud_rate: 256000
  parity: NONE
  stop_bits: 1

# ld2450 configuration
ld2450:
  id: ld2450_radar
  uart_id: uart_ld2450
  throttle: 1000ms

Configuration variables:

• id (Optional, config-id): Manually specify the ID for the ld2450 component if you need multiple components.
• uart_id (Optional, config-id): Manually specify the ID of the UART Component if you want to use multiple UART buses.
• throttle (Optional, int): Time in milliseconds to control the rate of data updates. Defaults to 1000ms.

Binary Sensor

The ld2450 binary sensor offers presence states for the targets.

binary_sensor:
  - platform: ld2450
    ld2450_id: ld2450_radar
    has_target:
      name: Presence
    has_moving_target:
      name: Moving Target
    has_still_target:
      name: Still Target

Configuration variables:

• ld2450_id (Optional, config-id): Manually specify the ID for the ld2450 component if you are using multiple components.
• has_target (Optional): True if either target is still or in motion. All options from Binary Sensor.
• has_moving_target (Optional): True if a moving target is detected. All options from Binary Sensor.
• has_still_target (Optional): True if a still target is detected. All options from Binary Sensor.

Sensor

The ld2450 sensor provides information about the states and attributes of detected targets.

sensor:
  - platform: ld2450
    ld2450_id: ld2450_radar
    target_count:
      name: Presence Target Count
    still_target_count:
      name: Still Target Count
    moving_target_count:
      name: Moving Target Count
    target_1:
      x:
        name: Target-1 X
      y:
        name: Target-1 Y
      speed:
        name: Target-1 Speed
      angle:
        name: Target-1 Angle
      distance:
        name: Target-1 Distance
      resolution:
        name: Target-1 Resolution
    target_2:
      x:
        name: Target-2 X
      y:
        name: Target-2 Y
      speed:
        name: Target-2 Speed
      angle:
        name: Target-2 Angle
      distance:
        name: Target-2 Distance
      resolution:
        name: Target-2 Resolution
    target_3:
      x:
        name: Target-3 X
      y:
        name: Target-3 Y
      speed:
        name: Target-3 Speed
      angle:
        name: Target-3 Angle
      distance:
        name: Target-3 Distance
      resolution:
        name: Target-3 Resolution
    zone_1:
      target_count:
        name: Zone-1 All Target Count
      still_target_count:
        name: Zone-1 Still Target Count
      moving_target_count:
        name: Zone-1 Moving Target Count
    zone_2:
      target_count:
        name: Zone-2 All Target Count
      still_target_count:
        name: Zone-2 Still Target Count
      moving_target_count:
        name: Zone-2 Moving Target Count
    zone_3:
      target_count:
        name: Zone-3 All Target Count
      still_target_count:
        name: Zone-3 Still Target Count
      moving_target_count:
        name: Zone-3 Moving Target Count

Configuration variables:

• ld2450_id (Optional, config-id): Manually specify the ID for the ld2450 component if you are using multiple components.
• target_count (Optional, int): Total targets detected, stationary or moving. All options from Sensor.
• still_target_count (Optional, int): Count of stationary targets. All options from Sensor.
• moving_target_count (Optional, int): Count of moving targets. All options from Sensor.
• target_N (Optional): Details about the targets (N = 1 to 3). A maximum of 3 targets can be detected: target_1, target_2, target_3.
  • x (Optional, int): Distance in millimeters (mm) of the target from the ld2450 sensor along the X-axis. All options from Sensor.
  • y (Optional, int): Distance in millimeters (mm) of the target from the ld2450 sensor in the Y direction. All options from Sensor.
  • speed (Optional, int): Speed of the moving target in mm/s. All options from Sensor.
  • angle (Optional, int): Angle of target in degrees (°) relative to the ld2450 sensor. All options from Sensor.
  • distance (Optional, int): Distance in millimeters (mm) of the target from the ld2450 sensor. All options from Sensor.
  • resolution (Optional, int): The ld2450 target detection range resolution in millimeters (mm). All options from Sensor.
• zone_N (Optional): Target count details in the defined zones (N = 1 to 3). A maximum of 3 zones: zone_1, zone_2, zone_3.
  • target_count (Optional, int): Total targets detected in the zone, stationary or moving. All options from Sensor.
  • still_target_count (Optional, int): Count of stationary targets in the zone. All options from Sensor.
  • moving_target_count (Optional, int): Count of moving targets in the zone. All options from Sensor.

Switch

The ld2450 switch allows you to control your ld2450 Bluetooth and Multi/Single Target Tracking.

switch:
  - platform: ld2450
    ld2450_id: ld2450_radar
    bluetooth:
      name: "Bluetooth"
    multi_target:
      name: "Multi Target Tracking"

Configuration variables:

• ld2450_id (Optional, config-id): Manually specify the ID for the ld2450 component if you are using multiple components.
• bluetooth (Optional): Turn on/off the Bluetooth adapter. Defaults to true. All options from Switch.
• multi_target (Optional): Turn on/off the Multi Target Tracking option. All options from Switch.

Number

The ld2450 number allows you to control the presence timeout and zone area configuration of your ld2450.

number:
  - platform: ld2450
    ld2450_id: ld2450_radar
    presence_timeout:
      name: "Timeout"
    zone_1:
      x1:
        name: Zone-1 X1
      y1:
        name: Zone-1 Y1
      x2:
        name: Zone-1 X2
      y2:
        name: Zone-1 Y2
    zone_2:
      x1:
        name: Zone-2 X1
      y1:
        name: Zone-2 Y1
      x2:
        name: Zone-2 X2
      y2:
        name: Zone-2 Y2
    zone_3:
      x1:
        name: Zone-3 X1
      y1:
        name: Zone-3 Y1
      x2:
        name: Zone-3 X2
      y2:
        name: Zone-3 Y2

Configuration variables:

• ld2450_id (Optional, config-id): Manually specify the ID for the ld2450 component if you are using multiple components.
• presence_timeout (Required, int): The duration, in seconds, for which the presence states will persist after detection is cleared. Default is 5 seconds. All options from Number.
• zone_N (Optional): Configuration of the zone (N = 1 to 3), defined as a rectangular area. A maximum of 3 Zones can be configured: zone_1, zone_2, zone_3. The zone coordinates consist of x1 y1 (Near-Left) to x2 y2 (Far-Right) relative to the ld2450 sensor.
  • x1 (Optional, int): Start X coordinate in millimeters (mm) of the zone from the ld2450 sensor along the X-axis. All options from Number.
  • y1 (Optional, int): Start Y coordinate in millimeters (mm) of the zone from the ld2450 sensor along the Y-axis. All options from Number.
  • x2 (Optional, int): End X coordinate in millimeters (mm) of the zone from the ld2450 sensor along the X-axis. All options from Number.
  • y2 (Optional, int): End Y coordinate in millimeters (mm) of the zone from the ld2450 sensor along the Y-axis. All options from Number.

Button

The ld2450 button allows you to perform restart and factory reset actions on your ld2450.

button:
  - platform: ld2450
    ld2450_id: ld2450_radar
    factory_reset:
      name: "LD2450 Factory Reset"
    restart:
      name: "LD2450 Restart"

Configuration variables:

• ld2450_id (Optional, config-id): Manually specify the ID for the ld2450 component if you are using multiple components.
• factory_reset (Optional): Resets the ld2450 to its default values. All options from Button.
• restart (Optional): Restarts the ld2450 device. All options from Button.
  

Text Sensor

The ld2450 text sensor allows you to get information about your ld2450.

text_sensor:
  - platform: ld2450
    ld2450_id: ld2450_radar
    version:
      name: "LD2450 Firmware"
    mac_address:
      name: "LD2450 BT MAC"
    target_1:
      direction:
        name: "Target-1 Direction"
    target_2:
      direction:
        name: "Target-2 Direction"
    target_3:
      direction:
        name: "Target-3 Direction"

Configuration variables:

• ld2450_id (Optional, config-id): Manually specify the ID for the ld2450 component if you are using multiple components.
• version (Optional): The ld2450 firmware version. All options from Text Sensor.
• mac_address (Optional): The ld2450 Bluetooth MAC address. All options from Text Sensor.
• target_N (Optional): Targets (N = 1 to 3). A maximum of 3 targets can be detected: target_1, target_2, target_3.
  • direction (Optional): Direction of the target relative to the ld2450 sensor. Possible values are: Stationary, Moving away, Coming closer, NA. All options from Sensor.

Select

The ld2450 select allows you to control baud rate and zone type of your ld2450.

select:
  - platform: ld2450
    ld2450_id: ld2450_radar
    baud_rate:
      name: "Baud rate"
    zone_type:
      name: "Zone Type"

Configuration variables:

• ld2450_id (Optional, config-id): Manually specify the ID for the ld2450 component if you are using multiple components.
• baud_rate (Optional): Control the UART serial port baud rate. Defaults to 256000. Once changed, all sensors will stop working until a fresh install with an updated UART Component configuration. All options from Select.
• zone_type (Optional): Control the Zone detection modes. It can be set to Disabled, Detection, or Filter. Selecting the Disabled option will disable zone area detection. Detection mode is used to detect only targets in the specified area, while Filter mode can be used to exclude an area from detection. All options from Select.

 

(Copied and adapted from hareeshmu)

Bluetooth Tracking with MTR-1

Please refer to the general Bluetooth tracking article

General Tips

MTR-1 PCB with Sensors

Light Sensor (LTR-390UV)

When mounting the MTR-1 be sure to position the device so that the large hole at the top is not covered. This allows more light to enter and will ensure better accuracy

The onboard RGB LED will trigger the light sensor. Be cognizant of this when making automations based on light/LUX.

Mounting

LD2450 Performance and Electrical Parameters

LD2450 Datasheet

Expanding Horizontal Distance as Target Moves Further Away

The detection is 60 degrees from center on each side. So it is a cone when you are close to it but at 4 feet away to the side you might not be visible. If you are further from the wall it can detect further in a horizontal distance.

Increased ESP Temperature

If you are experiencing higher than normal ESP temperatures ~140+ degrees F then changing the wifi power save mode option might help decrease the temperature. Here is the link to the ESPHome WiFi Component Power Save Mode.

Power Save Mode
The WiFi interface of all ESPs offer three power save modes to reduce the amount of power spent on WiFi. While some options can reduce the power usage of the ESP, they generally also decrease the reliability of the WiFi connection, with frequent disconnections from the router in the highest power saving mode.

NONE (least power saving, Default for ESP8266)
LIGHT (Default for ESP32)
HIGH (most power saving)

wifi:
  # ...
  power_save_mode: none

The code above can be added to the devices .yaml through the ESPHome addon edit button.

(Thank you for the suggestion, Brian!)
(Referenced from ESPHome website)

How To Change The Update Frequency Of Sensors/Force Update

How To Change The Update Frequency Of Sensors/Force Update

Zone Configuration Using Home Assistant

Disclaimer: Currently having issues saving the zones after restarting the device when using this method. It is recommended to use the HLKRadarTool App Zone Configuration Guide.

You will need to have HACS and Plotly installed to make it easier to visualize the zones. If you already have HACS then you can skip Step 1. Same for Step 2 if you have Plotly.

1. Install HACS
2. Install Plotly
3. Copy the code below and add a Home Assistant card to visualize your zones. You will need to change all of the sensor_apollo_mtr_1 entity IDs to match your MTR-1 device. This can be done quickly by using a code editor or ChatGPT.

type: custom:plotly-graph
title: Target Positions
refresh_interval: 1
hours_to_show: current_day
layout:
  height: 230
  margin:
    l: 50
    r: 20
    t: 20
    b: 40
  showlegend: true
  xaxis:
    dtick: 1000
    gridcolor: RGBA(200,200,200,0.15)
    zerolinecolor: RGBA(200,200,200,0.15)
    type: number
    fixedrange: true
    range:
      - 4000
      - -4000
  yaxis:
    dtick: 1000
    gridcolor: RGBA(200,200,200,0.15)
    zerolinecolor: RGBA(200,200,200,0.15)
    scaleanchor: x
    scaleratio: 1
    fixedrange: true
    range:
      - 7500
      - 0
entities:
  - entity: ''
    name: Target1
    marker:
      size: 12
    line:
      shape: spline
      width: 5
    x:
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_1_x"].state
    'y':
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_1_y"].state
  - entity: ''
    name: Target2
    marker:
      size: 12
    line:
      shape: spline
      width: 5
    x:
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_2_x"].state
    'y':
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_2_y"].state
  - entity: ''
    name: Target3
    marker:
      size: 12
    line:
      shape: spline
      width: 5
    x:
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_3_x"].state
    'y':
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_3_y"].state
  - entity: ''
    name: Zone1
    mode: lines
    fill: toself
    fillcolor: RGBA(20,200,0,0.06)
    line:
      color: RGBA(20,200,0,0.2)
      shape: line
      width: 2
    x:
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x1"].state
    'y':
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y1"].state
  - entity: ''
    name: Zone2
    mode: lines
    fill: toself
    fillcolor: RGBA(200,0,255,0.06)
    line:
      color: RGBA(200,0,255,0.2)
      shape: line
      width: 2
    x:
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_x1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_x1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_x2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_x2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_x1"].state
    'y':
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_y1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_y2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_y2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_y1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_2_y1"].state
  - entity: ''
    name: Zone3
    mode: lines
    fill: toself
    fillcolor: RGBA(200,120,55,0.06)
    line:
      color: RGBA(200,120,55,0.2)
      shape: line
      width: 2
    x:
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_x1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_x1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_x2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_x2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_x1"].state
    'y':
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_y1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_y2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_y2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_y1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_3_y1"].state
  - entity: ''
    name: Coverage
    mode: lines
    fill: tonexty
    fillcolor: rgba(168, 216, 234, 0.15)
    line:
      shape: line
      width: 1
      dash: dot
    x:
      - 0
      - $ex 7500 * Math.sin((2 * Math.PI)/360 * 60)
      - 4500
      - 4000
      - 3000
      - 2000
      - 1000
      - 0
      - -1000
      - -2000
      - -3000
      - -4000
      - -4500
      - $ex -7500 * Math.sin((2 * Math.PI)/360 * 60)
      - 0
    'y':
      - 0
      - $ex 7500 * Math.cos((2 * Math.PI)/360 * 60)
      - $ex Math.sqrt( 7500**2 - 4500**2 )
      - $ex Math.sqrt( 7500**2 - 4000**2 )
      - $ex Math.sqrt( 7500**2 - 3000**2 )
      - $ex Math.sqrt( 7500**2 - 2000**2 )
      - $ex Math.sqrt( 7500**2 - 1000**2 )
      - 7500
      - $ex Math.sqrt( 7500**2 - 1000**2 )
      - $ex Math.sqrt( 7500**2 - 2000**2 )
      - $ex Math.sqrt( 7500**2 - 3000**2 )
      - $ex Math.sqrt( 7500**2 - 4000**2 )
      - $ex Math.sqrt( 7500**2 - 4500**2 )
      - $ex 7500 * Math.cos((2 * Math.PI)/360 * 60)
      - 0
raw_plotly_config: true

4. Open Home Assistant and select Settings

5. Select Devices and services

6. Select ESPHome

7. Select your MTR-1 device

8. Scroll down to the Configuration section and you should see Zone 1-3 X and Y boxes.
- LD2450 Bluetooth can be turned on if you would like to use the HLKRadarTool App to configure the zones.
- Multi Target Tracking toggled on helps it detect up to three targets better (still trying to figure out "how" this works).
- Zone Type allows you to select Disabled, Detection, or Filter.
  - Disabled: Disable zone area detection
  - Detection: Only detects targets in the specified zone
  - Filter: Excludes a zone from detection

9. Now we can input our Zone 1-3 X and Y values to make our zones. Using the visual card from above, we can walk, sit, or stand in the area where we want to create a detection or non-detection zone. Input values for X are -7000 mm to 7000 mm, and the Y values are 0 mm to 7000 mm.

10. If you use the imperial system (Freedom Units) then you will need to do this step. Metric users can skip this. For the targets to show up correctly we first need to update the Target 1-3 X and Y measurements from inches (in) to millimeters (mm). Find Target 1-3 X and Y under the Sensors section and select them. You will want to update all 3 targets X and Y values.

11. Select the Settings cog in the top right.

12. Change the unit of measurement to mm and select Update.

13. Now you should see targets on the card.

14. Now we can make zones around the targets where you want to Detect presence or filter them out!
15. If your zone configurations are not saved when restarting the device then try turning on the LD2450 Bluetooth for a few seconds and then turning it off again. You can also try to toggle on Multi Target Radar. This should wake up the mmWave module and retrieve your saved zones. (Thanks to our Discord members for helping figure this out)

Zone Configuration Using HLKRadarTool App

1. Download the HLKRadarTool app for either Android or Apple
2. Ensure the mmWave radar you want to configure has Radar Control Bluetooth turned on. Home Assistant > Settings > Devices & services > ESPHome Devices > Select Device > Scroll down and toggle on Radar Control Bluetooth

3. Open the HLKRadarTool App and select your device

4. Select Set in the top right

5. Select Close Detection, Area Detection, or Area Filtering. Then toggle Area 1, 2, or 3 and you will see a box with the corresponding number pop up. Then you can press and hold the box to move it and resize it. When your zones are set then select Submit and you should see Setup successfully!
  - Close Detection: Disable zone area detection
  - Area Detection: Only detects targets in the specified zone
  - Area Filtering: Excludes a zone from detection

6. If set up correctly the zones should be saved as different colors. You can double-check that the settings are saved by looking at your HA entities (picture below). You should be all set! See below for a nice Home Assistant visualizer card.

7. Copy the code below and add a Home Assistant card to visualize your zones. You will need to change all of the sensor_apollo_mtr_1 entity IDs to match your MTR-1 device. This can be done quickly by using a code editor or ChatGPT.

type: custom:plotly-graph
title: Target Positions
refresh_interval: 1
hours_to_show: current_day
layout:
  height: 230
  margin:
    l: 50
    r: 20
    t: 20
    b: 40
  showlegend: true
  xaxis:
    dtick: 1000
    gridcolor: RGBA(200,200,200,0.15)
    zerolinecolor: RGBA(200,200,200,0.15)
    type: number
    fixedrange: true
    range:
      - 4000
      - -4000
  yaxis:
    dtick: 1000
    gridcolor: RGBA(200,200,200,0.15)
    zerolinecolor: RGBA(200,200,200,0.15)
    scaleanchor: x
    scaleratio: 1
    fixedrange: true
    range:
      - 7500
      - 0
entities:
  - entity: ''
    name: Target1
    marker:
      size: 12
    line:
      shape: spline
      width: 5
    x:
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_1_x"].state
    'y':
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_1_y"].state
  - entity: ''
    name: Target2
    marker:
      size: 12
    line:
      shape: spline
      width: 5
    x:
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_2_x"].state
    'y':
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_2_y"].state
  - entity: ''
    name: Target3
    marker:
      size: 12
    line:
      shape: spline
      width: 5
    x:
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_3_x"].state
    'y':
      - $ex hass.states["sensor.apollo_mtr_1_982da4_target_3_y"].state
  - entity: ''
    name: Zone1
    mode: lines
    fill: toself
    fillcolor: RGBA(20,200,0,0.06)
    line:
      color: RGBA(20,200,0,0.2)
      shape: line
      width: 2
    x:
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_x1"].state
    'y':
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y2"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y1"].state
      - $ex hass.states["number.apollo_mtr_1_982da4_zone_1_y1"].state
  - entity: ''
    name: Zone2
    mode: lines
    fill: toself
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      - $ex Math.sqrt( 7500**2 - 4000**2 )
      - $ex Math.sqrt( 7500**2 - 3000**2 )
      - $ex Math.sqrt( 7500**2 - 2000**2 )
      - $ex Math.sqrt( 7500**2 - 1000**2 )
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      - $ex Math.sqrt( 7500**2 - 3000**2 )
      - $ex Math.sqrt( 7500**2 - 4000**2 )
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raw_plotly_config: true