===============================================================================
  PHISUALIZE IT: THE SCIENCE - TURNING SIGNALS INTO MEANING
  Detailed Scientific Analysis of 10 Invisible Phenomena
===============================================================================

This document explains the physics, mathematics, and signal processing behind
each phenomenon detected by PHISUALIZE IT!.

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1. MECHANICAL VIBRATIONS (IMU - BMI270 Accelerometer)
===============================================================================

PHYSICAL PRINCIPLE:
Mechanical vibrations are oscillatory movements measured as acceleration
variations. The device detects vibrations through dynamic changes in the
acceleration vector, removing static gravity component.

SENSOR: BMI270 (3-axis accelerometer)
SAMPLING RATE: 90 Hz
UNIT: g RMS (root-mean-square acceleration, where 1g = 9.81 m/s²)

SIGNAL PROCESSING CHAIN:

1) Raw Acceleration (3D vector):
   a_total(t) = [ax(t), ay(t), az(t)]
   
   Where a_total includes both:
   - Static component: gravity (~1g constant)
   - Dynamic component: vibrations (AC signal)

2) High-Pass Filtering (gravity removal):
   Each axis filtered independently with α = 0.88:
   
   y[n] = (x[n] - x[n-1]) + α·y[n-1]
   
   Cutoff frequency: fc ≈ (1-α)·fs/(2π) = 0.12×90/6.28 ≈ 1.7 Hz
   
   This removes DC (gravity) and keeps AC (vibrations > 1.7 Hz)

3) Magnitude Vector (orientation-invariant):
   ||a_dyn(t)|| = √(ax_filt² + ay_filt² + az_filt²)
   
   Result: scalar representing total vibrational energy regardless of direction

4) FFT Analysis (N=128 samples):
   Window: 128/90 = 1.42 seconds
   Resolution: 90/128 ≈ 0.7 Hz per bin
   
   Spectral amplitude: A(f) = |FFT{a_dyn(t)}|
   
   Dominant frequency: f_dom = argmax(A(f))
   Dominant amplitude: A_max = max(A(f))

5) Normalization for display:
   Energy_normalized = A_max / 0.10g
   Brightness = 30 + (Energy_normalized × 225)
   
   Reference: 0.10g = strong vibration (walking, transport)

TYPICAL VALUES:
- Baseline noise: 0.001 - 0.005 g
- Detectable vibrations: > 0.01 g
- Strong impact: > 0.5 g

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2. AUDIO SPECTRUM (PDM Microphone MP34DT06JTR)
===============================================================================

PHYSICAL PRINCIPLE:
Sound waves are pressure variations in air detected by microphone membrane.
Structured sounds (voice, music) have harmonic content, while electromagnetic
interference produces flat/noisy spectrum.

SENSOR: MP34DT06JTR (PDM digital microphone)
SAMPLING RATE: 16 kHz
UNIT: Arbitrary spectral amplitude (normalized 0-255 per band)

SIGNAL PROCESSING CHAIN:

1) PDM to PCM Conversion:
   Raw PDM bitstream → 16-bit signed integers (-32768 to +32767)

2) High-Pass Filtering (DC removal):
   α = 0.962, fc ≈ 1 Hz @ fs=16kHz
   
   Removes DC offset and thermal drift without affecting 50 Hz mains
   interference (relevant for EMI detection)

3) Peak Detection (silence threshold):
   Peak = max(|sample[i]|) for i ∈ [0, 127]
   
   If Peak < 15 → Digital silence (below ADC noise floor ~-90 dBFS)

4) FFT Analysis (N=128 samples):
   Window: 128/16000 = 8 ms
   Resolution: 16000/128 = 125 Hz per bin
   
   Magnitude spectrum: |X(f)| = √(Re² + Im²)

5) 8-Band Harmonic Decomposition:
   Bands aligned with 125 Hz resolution:
   
   Band 0: 0-125 Hz     (mains 50 Hz + low fundamentals)
   Band 1: 125-250 Hz   (H1 = 125 Hz)
   Band 2: 250-375 Hz   (H2 = 250 Hz)
   Band 3: 375-500 Hz   (H3 = 375 Hz)
   Band 4: 500-625 Hz   (H4 = 500 Hz)
   Band 5: 625-750 Hz   (H5 = 625 Hz)
   Band 6: 750-875 Hz   (H6 = 750 Hz)
   Band 7: 875-1000 Hz  (H7 = 875 Hz)
   
   Each band: Average magnitude of corresponding FFT bins
   Normalization: FIXED scale (50.0) + logarithmic compression

6) Color Mapping:
   Rainbow spectrum: Red → Orange → Yellow → Green → Cyan → Blue → Magenta → Violet
   Each band gets unique color representing its frequency range

TYPICAL VALUES:
- Silence: < 0.002 FFT amplitude
- Speech: 0.05 - 0.15 FFT amplitude
- Loud music: > 0.20 FFT amplitude

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3. EMI INTERFERENCE (Same PDM Microphone - Dual Mode)
===============================================================================

PHYSICAL PRINCIPLE:
Electromagnetic interference (switching power supplies, motors, radio) induces
currents in microphone circuitry, producing acoustic artifacts with flat
spectrum (white/pink noise) unlike structured audio harmonics.

INNOVATION: Single sensor dual classification via spectral analysis

CLASSIFICATION METHOD - Spectral Flatness Measure (SFM):

SFM = Geometric_Mean / Arithmetic_Mean

Mathematically:
         exp(1/N × Σ log|X(f)|)
SFM = ─────────────────────────
            1/N × Σ|X(f)|

Where N = number of valid frequency bins

INTERPRETATION:
- SFM → 1.0: Flat spectrum (all frequencies equal) = White noise = EMI
- SFM → 0.0: Peaked spectrum (few harmonics) = Structured sound = Audio

DECISION THRESHOLDS (with hysteresis):
1) If SFM > 0.45 → EMI (electromagnetic interference)
2) If SFM < 0.40 → AUDIO (voice, music)
3) If 0.40 ≤ SFM ≤ 0.45 → Keep previous state (prevents oscillation)

AMPLITUDE THRESHOLD:
If amplitude < 0.002 → Classify as EMI (thermal noise floor)

VISUALIZATION:
EMI displays as colored "snow" - random pixels weighted by spectral content
Number of pixels proportional to total spectral energy

TYPICAL EMI SOURCES:
- Switching power supplies (50-150 kHz fundamental, audio harmonics)
- Electric motors (brush noise)
- Wi-Fi/Bluetooth (2.4 GHz with audio-range modulation artifacts)
- LED drivers (PWM artifacts)

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4. MAGNETIC FIELD (BMM150 Magnetometer)
===============================================================================

PHYSICAL PRINCIPLE:
Earth's magnetic field is ~25-65 µT depending on location. Ferromagnetic
objects (iron, steel) locally distort this field. Movement of permanent
magnets or current-carrying conductors creates detectable variations.

SENSOR: BMM150 (3-axis magnetometer)
SAMPLING RATE: ~50 Hz
UNIT: µT (microtesla)

MEASUREMENT:

1) Raw 3D Magnetic Vector:
   B(t) = [Bx(t), By(t), Bz(t)]  in µT

2) Total Field Magnitude:
   ||B|| = √(Bx² + By² + Bz²)

3) Baseline Calibration (at startup):
   B_baseline = ⟨||B||⟩  averaged over 100 samples (~2 seconds)
   
   Typical values:
   - Equator: ~25-30 µT
   - Mid-latitudes: ~45-55 µT
   - Poles: ~60-65 µT

4) Variation Detection:
   ΔB(t) = ||B(t)|| - B_baseline
   
   Smoothing: Low-pass filter with α=0.4

5) Normalization for Display:
   Range: 0-50 µT (covers typical disturbances)
   
   Energy_normalized = |ΔB| / 50 µT
   Brightness = 30 + (Energy_normalized × 225)

6) Compass Angle (2D projection):
   θ = atan2(By, Bx)  [radians]
   
   Converted to degrees: θ_deg = θ × 180/π
   If θ_deg < 0: θ_deg += 360°

TYPICAL DISTURBANCES:
- Smartphone: +50 µT at 10 cm
- Neodymium magnet: +500 µT at 5 cm
- Steel beam: ±20 µT
- Electric current (1A): ~2 µT at 10 cm (Biot-Savart law)

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5. INFRARED PROXIMITY (APDS9960)
===============================================================================

PHYSICAL PRINCIPLE:
Infrared LED emits light (~940 nm), photodiode measures reflected intensity.
Closer objects reflect more light. Intensity follows inverse square law.

SENSOR: APDS9960 (proximity + gesture + RGB/ambient light)
SAMPLING RATE: ~10 Hz
UNIT: Arbitrary units (0-255)

PHYSICS:

1) IR Reflectance (simplified):
   I_received ∝ (ρ × P_emitted) / r²
   
   Where:
   - ρ = surface reflectivity (albedo)
   - P_emitted = IR LED power
   - r = distance to object

2) Sensor Output:
   Proximity_raw ∈ [0, 255]
   
   0 = No object detected (>25 cm)
   255 = Object very close (<3 cm)

3) Exponential Smoothing:
   Proximity_filtered = α × Proximity_raw + (1-α) × Proximity_filtered_prev
   
   With α = 0.2 (moderately fast response)

4) Normalization:
   Proximity_normalized = Proximity_filtered / 255.0  ∈ [0, 1]

VISUALIZATION LOGIC:
- Central IR "hotspot" (4 pixels)
- If close (>0.3): Expand cross pattern (8 more pixels)
- If far (<0.5): Expanding circles (inverse distance)
- If near (>0.5): Multiple concentric circles (proximity levels)

TYPICAL DISTANCES:
- Far: 0-30 (>15 cm)
- Medium: 30-100 (5-15 cm)
- Near: 100-200 (3-5 cm)
- Very near: 200-255 (<3 cm)

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6. GPS SATELLITES (GY-GPS6MV2 NEO-6M)
===============================================================================

PHYSICAL PRINCIPLE:
GPS satellites (24+ in orbit, ~20,000 km altitude) broadcast L1 signals
(1575.42 MHz). Receiver measures time-of-flight from 4+ satellites to
triangulate position via trilateration.

SENSOR: NEO-6M GPS module (u-blox)
SAMPLING RATE: 1 Hz (NMEA sentences)
UNIT: Latitude/Longitude (degrees), Altitude (meters), Satellites (count)

POSITIONING MATHEMATICS:

1) Time-of-Flight Measurement:
   Distance to satellite i:
   d_i = c × (t_receive - t_transmit)
   
   Where c = 299,792,458 m/s (speed of light)

2) Trilateration (4 satellites minimum):
   System of equations:
   (x - x_i)² + (y - y_i)² + (z - z_i)² = d_i²
   
   For i = 1, 2, 3, 4 (4 unknowns: x, y, z, clock_error)

3) NMEA GPGGA Parsing:
   Example: $GPGGA,123519,4807.038,N,01131.000,E,1,08,0.9,545.4,M,46.9,M,,*47
   
   Fields:
   - Time: 12:35:19 UTC
   - Latitude: 48°07.038' N → 48 + 07.038/60 = 48.1173°
   - Longitude: 11°31.000' E → 11 + 31.000/60 = 11.5167°
   - Fix quality: 1 (GPS fix)
   - Satellites: 08
   - Altitude: 545.4 m

4) Signal Quality Metric:
   Quality = min(Satellites / 12, 1.0)
   
   Optimal: 12 satellites (100% quality)
   Minimum for 3D fix: 4 satellites
   Typical urban: 6-8 satellites

5) Timeout Handling:
   If no NMEA data for 2 seconds → Satellites = 0
   Triggers "searching" animation (4 rotating points)

VISUALIZATION:
- 12 positions on matrix (4 center + 8 cardinal/intercardinal)
- Active satellites: Bright
- Inactive satellites: Dim baseline
- No fix: Rotating search animation

ACCURACY:
- Open sky: ±3-5 meters (CEP)
- Urban canyon: ±10-20 meters
- Indoor: No fix (signal too weak)

===============================================================================
7. BLUETOOTH (BLE - Nordic nRF52840)
===============================================================================

PHYSICAL PRINCIPLE:
Bluetooth Low Energy uses 2.4 GHz ISM band (2400-2483.5 MHz) with adaptive
frequency hopping. Signal strength (RSSI) indicates link quality and
approximate distance.

SENSOR: nRF52840 (built-in BLE radio)
SAMPLING RATE: Continuous when connected
UNIT: dBm (decibel-milliwatts)

RADIO PHYSICS:

1) Path Loss (Friis Transmission Equation):
   RSSI = P_tx + G_tx + G_rx - PL
   
   Where:
   - P_tx = Transmit power (typically 0 dBm = 1 mW)
   - G_tx, G_rx = Antenna gains
   - PL = Path loss (free space):
   
   PL = 20×log₁₀(d) + 20×log₁₀(f) + 20×log₁₀(4π/c)
   
   For d in meters, f in Hz, c = speed of light

2) RSSI Measurement:
   Typical range: -100 dBm (far) to -30 dBm (very close)
   
   Distance estimation (rough):
   d ≈ 10^((P_tx - RSSI)/(10×n))
   
   Where n = path loss exponent (2.0 in free space, 2.5-4.0 indoors)

3) Normalization:
   RSSI_normalized = (RSSI + 100) / 70
   
   Maps [-100, -30] dBm → [0.0, 1.0]
   
   Clamped: min(max(RSSI_normalized, 0.0), 1.0)

VISUALIZATION:
- No connection: Pulsing circle (breathing animation)
- Connected: Concentric circles (1-4 rings based on signal strength)
- Central point: Always visible (4 pixels)

TYPICAL RSSI VALUES:
- -30 dBm: Device touching (0-10 cm)
- -50 dBm: Very close (10-50 cm)
- -70 dBm: Normal range (1-5 m)
- -90 dBm: Weak signal (5-10 m)
- -100 dBm: Connection limit

DATA STREAMING:
13 BLE characteristics transmit all sensor data to ESP32 touchscreen
- IMU magnitude
- Magnetic delta
- Proximity
- Pressure
- Audio/EMI frequencies (Hz)
- Temperature
- Humidity
- GPS coordinates
- Compass angle
- 8-band spectrum

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8. ATMOSPHERIC PRESSURE (LPS22HB Barometer)
===============================================================================

PHYSICAL PRINCIPLE:
Atmospheric pressure is the weight of air column above a point. Variations
indicate weather changes, altitude differences, or local pressure systems.

SENSOR: LPS22HB (MEMS barometer)
SAMPLING RATE: ~0.5 Hz (slow, stable readings)
UNIT: hPa (hectopascal) = mbar (millibar)

MEASUREMENT:

1) Raw Pressure Reading:
   Arduino library returns kPa → Multiply by 10 for hPa
   
   P_raw [kPa] × 10 = P [hPa]

2) Baseline Calibration:
   P_baseline = ⟨P⟩  averaged over 100 samples (~200 seconds)
   
   Sea level standard: 1013.25 hPa
   Typical range: 980-1040 hPa

3) Pressure Variation:
   ΔP(t) = P(t) - P_baseline

4) Exponential Smoothing:
   α = 0.05 (very slow, ~2 second time constant)
   
   P_smooth = α × P_raw + (1-α) × P_smooth_prev

5) Altitude Estimation (barometric formula):
   Δh ≈ -8.3 × ΔP  [meters per hPa]
   
   More accurate formula:
   h = 44330 × (1 - (P/P₀)^0.1903)
   
   Where P₀ = 1013.25 hPa (sea level standard)

METEOROLOGICAL INTERPRETATION:
- ΔP > +5 hPa: High pressure system (clear weather)
- ΔP < -5 hPa: Low pressure system (storms)
- Rapid drop: Weather front approaching
- Stable: Settled conditions

TYPICAL VARIATIONS:
- Weather systems: ±10-30 hPa over days
- Building floors: -1.2 hPa per 10 meters
- Breathing on sensor: +0.1-0.5 hPa transient

===============================================================================
9. TEMPERATURE (HS3003)
===============================================================================

PHYSICAL PRINCIPLE:
Temperature measures thermal energy - average kinetic energy of molecules.
HS3003 uses resistance change in sensing element (thermistor or RTD).

SENSOR: HS3003 (integrated temp/humidity)
SAMPLING RATE: ~0.5 Hz
UNIT: °C (Celsius)
RANGE: -40°C to +85°C
ACCURACY: ±0.2°C typical

MEASUREMENT:

1) Raw Temperature Reading:
   T [°C] from sensor (already calibrated)

2) Baseline Calibration:
   T_baseline = ⟨T⟩  averaged over 50 samples (~100 seconds)

3) Temperature Variation:
   ΔT(t) = T(t) - T_baseline

4) Exponential Smoothing:
   α = 0.05 (slow response, matches thermal mass)

DERIVED METRICS:

A) Heat Index (feels-like temperature):
   Valid for T > 20°C and RH > 40%
   
   Simplified Steadman formula:
   HI = -8.785 + 1.611×T + 2.339×RH - 0.146×T×RH
        - 0.0123×T² - 0.0164×RH² + 0.00221×T²×RH
        + 0.000725×T×RH² - 0.00000358×T²×RH²
   
   Where T in °C, RH in %

B) Comfort Zone (ASHRAE):
   Comfortable: 18-24°C and 30-60% RH

TYPICAL VARIATIONS:
- Room temp: 18-25°C
- Body heat: +2-5°C near skin
- Breath: +10-20°C transient
- Sunlight: +5-15°C

===============================================================================
10. HUMIDITY (HS3003)
===============================================================================

PHYSICAL PRINCIPLE:
Relative humidity (RH) is ratio of actual water vapor pressure to saturation
vapor pressure at given temperature. Sensor uses capacitive element whose
dielectric constant changes with moisture absorption.

SENSOR: HS3003 (capacitive humidity sensor)
SAMPLING RATE: ~0.5 Hz
UNIT: % RH (relative humidity)
RANGE: 0-100% RH
ACCURACY: ±2% RH typical

MEASUREMENT:

1) Raw Humidity Reading:
   RH [%] from sensor

2) Baseline Calibration:
   RH_baseline = ⟨RH⟩  averaged over 50 samples

3) Humidity Variation:
   ΔRH(t) = RH(t) - RH_baseline

4) Exponential Smoothing:
   α = 0.1 (slightly faster than temperature, moisture diffuses quickly)

DERIVED METRICS:

A) Dew Point (Magnus-Tetens Formula):
   Temperature at which air becomes saturated (RH=100%)
   
   Constants: a = 17.27, b = 237.7°C
   
   α = ((a×T)/(b+T)) + ln(RH/100)
   
   T_dew = (b×α) / (a-α)
   
   Physical meaning: Water vapor condenses if cooled to T_dew

B) Absolute Humidity:
   Actual water vapor mass per volume
   
   AH = (RH × P_sat) / (R × T)
   
   Where:
   - P_sat = Saturation vapor pressure (function of T)
   - R = Gas constant
   - T in Kelvin

TYPICAL VALUES:
- Desert: 10-30% RH
- Comfortable: 30-60% RH
- Tropical: 70-90% RH
- Breath: +20-40% RH transient
- Steam: 100% RH

===============================================================================
COMPOSITE ENVIRONMENTAL (ENV) VISUALIZATION
===============================================================================

The ENV mode (mode 8) combines three environmental parameters into a single
PHI-symbol visualization:

1) ATMOSPHERIC PRESSURE → Circle radius
   Normalization: |ΔP| / 5 hPa → [0, 1]
   
   Visual: Circle expands/contracts with pressure changes

2) TEMPERATURE → Vertical bar color gradient
   Normalization: ΔT / 10°C → [-1, 1]
   
   Visual:
   - Cold (ΔT < -3°C): Blue gradient
   - Neutral (|ΔT| < 3°C): Purple/white
   - Hot (ΔT > +3°C): Red/orange gradient

3) HUMIDITY → Peripheral "cloud" density
   Normalization: |ΔRH| / 30% → [0, 1]
   
   Visual: Number of corner pixels (6-16) representing moisture

Combined Formula:
ENV_composite = 2.0×|ΔP| + 1.5×|ΔT| + 0.1×|ΔRH|

This creates a unified environmental "fingerprint" showing how all three
atmospheric parameters deviate from baseline simultaneously.

===============================================================================
VIBRO-ACOUSTIC COUPLING ANALYSIS
===============================================================================

PRINCIPLE:
Mechanical vibrations can couple with acoustic waves through structural
resonances. Examples: speaker cone, drum membrane, building structures.

COUPLING SCORE CALCULATION:

Given:
- f_IMU: Dominant vibration frequency (Hz)
- f_AUD: Dominant audio frequency (Hz)
- E_IMU: IMU spectral amplitude
- E_AUD: Audio spectral amplitude

1) Frequency Correlation:
   Measures how close the dominant frequencies are
   
   Δf = |f_IMU - f_AUD|
   
   C_freq = exp(-Δf / σ)
   
   Where σ = 100 Hz (typical phoneme bandwidth)
   
   Result: C_freq ∈ [0, 1]
   - 1.0 if frequencies match exactly
   - 0.37 if 100 Hz apart
   - 0.05 if 300 Hz apart

2) Energy Correlation:
   Measures if both signals have significant amplitude
   
   Normalize each amplitude:
   E_IMU_norm = E_IMU / 0.10  (reference: strong vibration)
   E_AUD_norm = E_AUD / 0.10  (reference: loud sound)
   
   Geometric mean:
   C_energy = √(E_IMU_norm × E_AUD_norm)
   
   Saturate at 1.0:
   C_energy = min(C_energy, 1.0)

3) Combined Coupling Score:
   Coupling = C_freq × C_energy  ∈ [0, 1]
   
   Interpretation:
   - 0.0: No coupling (different frequencies OR weak signals)
   - 0.5: Partial coupling
   - 1.0: Perfect coupling (same frequency, both strong)

PHYSICAL EXAMPLES:
- Speaker: High coupling (0.8-1.0)
- Hand clapping: Medium coupling (0.4-0.6)
- Typing + music: Low coupling (0.0-0.2)
- Machine vibration + motor hum: High coupling (0.7-0.9)

===============================================================================
BRIGHTNESS = REALITY PRINCIPLE
===============================================================================

Core Scientific Philosophy:

Every phenomenon's intensity maps LINEARLY to LED brightness:

Brightness = B_min + (Energy_normalized × (B_max - B_min))

Where:
- B_min = 30 (always-visible baseline)
- B_max = 255 (maximum LED intensity)
- Energy_normalized ∈ [0, 1] (phenomenon-specific normalization)

KEY PRINCIPLE: No artificial amplification or compression
- Weak signal → Dim LEDs
- Strong signal → Bright LEDs

This ensures scientific rigor: Visual brightness directly represents
physical intensity of detected phenomenon.

NORMALIZATION REFERENCES (per phenomenon):
1. IMU: 0.10g (strong vibration)
2. Audio: Fixed scale 50.0 (empirical calibration)
3. EMI: Same as audio
4. Magnetic: 50 µT (strong disturbance)
5. Proximity: 255 (sensor maximum)
6. GPS: 12 satellites (optimal)
7. BLE: 70 dBm range (-100 to -30)
8. Barometer: 5 hPa (significant variation)
9. Temperature: 10°C (noticeable change)
10. Humidity: 30% RH (significant change)

===============================================================================
SIGNAL PROCESSING SUMMARY
===============================================================================

COMMON TECHNIQUES USED:

1) High-Pass Filtering (DC removal):
   y[n] = (x[n] - x[n-1]) + α·y[n-1]
   
   Removes static components (gravity, DC offset, drift)

2) Low-Pass Filtering (smoothing):
   y[n] = α·x[n] + (1-α)·y[n-1]
   
   Reduces noise, creates visually stable display

3) Fast Fourier Transform (FFT):
   X(f) = Σ x(t)·e^(-j2πft)
   
   Converts time-domain to frequency-domain for spectral analysis

4) Hann Window (spectral leakage reduction):
   w(n) = 0.5 × (1 - cos(2πn/(N-1)))
   
   Applied before FFT to reduce side-lobes

5) Exponential Moving Average (EMA):
   y[n] = α·x[n] + (1-α)·y[n-1]
   
   Simple, efficient smoothing (α = smoothing factor)

===============================================================================
END OF SCIENTIFIC DOCUMENTATION
===============================================================================

This document provides the mathematical and physical foundation for
understanding how PHI-SPHERE transforms raw sensor data into meaningful
visualizations of invisible phenomena.

For implementation details, see source code comments in respective modules:
- PhiIMU.cpp: Vibration processing
- PhiAudio.cpp: Audio capture & silence detection
- PhiScientificModel.cpp: FFT analysis & Audio/EMI classification
- PhiMagnetometer.cpp: Magnetic field measurement
- PhiBarometer.cpp: Pressure sensing
- PhiHumidity.cpp: Temperature & humidity
- PhiProximity.cpp: IR detection
- PhiGPS.cpp: NMEA parsing
- PhiBLE.cpp: Wireless streaming
- PhiMatrixEngine.cpp: LED visualization algorithms

===============================================================================