Measurements

“Man started from measuring himself and finished weighing the stars.”

Measurement

Measurement is the process of determining the size, length, or amount of something. It is a fundamental aspect of physics and other sciences, as it allows us to quantify and compare physical quantities. Today we will discuss some of the most important measurements in the history of physics and describe the importance and details of making measurements.

Types of measurements in physics

• Direct measurements: Measuring a physical quantity directly using a measuring instrument, such as a ruler, thermometer, or voltmeter.

• Indirect measurements: Measuring a physical quantity indirectly by using other measurements and mathematical relationships, such as calculating the volume of a sphere from its diameter, or determining the speed of an object from its position and time.

Difference between accuracy and precision

• Accuracy is the degree to which a measurement is close to the true value of the quantity being measured. It is a measure of how well the measurement reflects the actual physical quantity.

• Precision is the degree to which repeated measurements of the same quantity give consistent results. It is a measure of the reproducibility and consistency of the measurement process.

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle
import matplotlib.gridspec as gridspec

# Set random seed for reproducibility
np.random.seed(42)

# Function to generate shots with different accuracy and precision patterns
def generate_shots(n_shots=50, center=(0, 0), 
                  accuracy_offset=(0, 0), 
                  precision_factor=1.0):
    """
    Generate simulated shots for target practice
    
    Parameters:
    - n_shots: number of shots to generate
    - center: true target center (x, y)
    - accuracy_offset: how far off the average is from true center (x, y)
    - precision_factor: standard deviation controlling spread (lower = better precision)
    """
    # Apply accuracy offset to the center
    biased_center = (center[0] + accuracy_offset[0], center[1] + accuracy_offset[1])
    
    # Generate random points around the biased center with given precision
    x = np.random.normal(biased_center[0], precision_factor, n_shots)
    y = np.random.normal(biased_center[1], precision_factor, n_shots)
    
    return x, y

# Create figure with subplots
plt.figure(figsize=(14, 10))
gs = gridspec.GridSpec(2, 2, width_ratios=[1, 1], height_ratios=[1, 1])

# Define scenarios to visualize
scenarios = [
    {"title": "High Accuracy, High Precision", 
     "accuracy_offset": (0.5, 0.5), 
     "precision_factor": 0.5},
    
    {"title": "Low Accuracy, High Precision", 
     "accuracy_offset": (3.0, -2.0), 
     "precision_factor": 0.5},
    
    {"title": "High Accuracy, Low Precision", 
     "accuracy_offset": (0.5, 0.5), 
     "precision_factor": 2.0},
    
    {"title": "Low Accuracy, Low Precision", 
     "accuracy_offset": (3.0, -2.0), 
     "precision_factor": 2.0}
]

# Draw target and shots for each scenario
for i, scenario in enumerate(scenarios):
    ax = plt.subplot(gs[i])
    
    # Draw target rings
    for r in [5, 4, 3, 2, 1]:
        circle = Circle((0, 0), r, fill=False, edgecolor='black', linewidth=1)
        ax.add_patch(circle)
    
    # Generate and plot shots
    x, y = generate_shots(
        n_shots=50, 
        center=(0, 0), 
        accuracy_offset=scenario["accuracy_offset"], 
        precision_factor=scenario["precision_factor"]
    )
    
    # Calculate mean position (center of mass) of shots
    mean_x, mean_y = np.mean(x), np.mean(y)
    
    # Plot individual shots
    ax.scatter(x, y, color='blue', alpha=0.6, s=30, label='Shots')
    
    # Plot true center and observed mean
    ax.scatter(0, 0, color='red', s=100, marker='+', label='True Center')
    ax.scatter(mean_x, mean_y, color='green', s=100, marker='x', label='Shot Mean')
    
    # Draw line from true center to mean (showing accuracy vector)
    ax.arrow(0, 0, mean_x, mean_y, color='red', width=0.05, head_width=0.3, alpha=0.7,
             length_includes_head=True)
    
    # Configure plot
    ax.set_xlim(-7, 7)
    ax.set_ylim(-7, 7)
    ax.set_aspect('equal')
    ax.grid(True, linestyle='--', alpha=0.7)
    ax.set_title(scenario["title"], fontsize=12)
    
    # Add accuracy and precision information
    accuracy_distance = np.sqrt(scenario["accuracy_offset"][0]**2 + scenario["accuracy_offset"][1]**2)
    precision_value = scenario["precision_factor"]
    
    # Calculate actual measured accuracy and precision from data
    actual_accuracy = np.sqrt(mean_x**2 + mean_y**2)  # Distance from true center to mean
    actual_precision = np.mean(np.sqrt((x - mean_x)**2 + (y - mean_y)**2))  # Average distance to mean
    
    info_text = f"Accuracy: {'High' if accuracy_distance < 2 else 'Low'}\nPrecision: {'High' if precision_value < 1 else 'Low'}"
    stats_text = f"Accuracy measure: {actual_accuracy:.2f}\nPrecision measure: {actual_precision:.2f}"
    
    ax.text(0.05, 0.95, info_text, transform=ax.transAxes, 
            verticalalignment='top', bbox={'boxstyle': 'round', 'facecolor': 'wheat', 'alpha': 0.5})
    ax.text(0.05, 0.80, stats_text, transform=ax.transAxes, 
            verticalalignment='top', bbox={'boxstyle': 'round', 'facecolor': 'lightblue', 'alpha': 0.5})

# Add a legend to the last subplot
handles, labels = ax.get_legend_handles_labels()
plt.figlegend(handles, labels, loc='lower center', ncol=3, bbox_to_anchor=(0.5, 0.05))

# Add overall title
plt.suptitle('Target Shooting: Visualizing Accuracy vs Precision', fontsize=16, y=0.98)
plt.subplots_adjust(bottom=0.15, wspace=0.2, hspace=0.3)


plt.tight_layout(rect=[0, 0.12, 1, 0.95])
plt.savefig('accuracy_vs_precision_target.png', dpi=300, bbox_inches='tight')
plt.show()

Why do we measure?

• To capture relations within the physical world

• To make predictions about the behavior of the natural world

• To understand the physical world

• To confirm and refine our understanding of the laws of physics

Measurement is essential in physics and other sciences because it allows us to describe and understand the physical world. Without measurement, we would not be able to quantify physical quantities, compare different objects or events, or make predictions about the behavior of the natural world.

flowchart TD
    A[Measurements] --> B[Mathematical Model]
    B --> C[Measurements]
    C --> D[Predictions about the future]

Units of measurement

To make meaningful measurements, we need to use standardized units of measurement (SI units). These units are defined by international agreements and are used by scientists around the world.

SI Units in physics include only seven units:

• meter \([m]\) for length

• kilogram \([kg]\) for mass

• second \([s]\) for time

• Kelvin \([K]\) for temperature

• ampere \([A]\) for electric current

• mole \([mol]\) for the amount of substance

• candela \([cd]\) for luminous intensity

All other units are derived from these base units. For example, the unit of force, the Newton (N), is derived from the base units of mass, length, and time.

Examples of derived units in physics include:

• Acceleration: \([\frac{m}{s^2}]\)

\[a = \frac{d v}{d t}\]

• Force: Newton \([N]=[\frac{kg\,m}{s^2}]\)

\[F = ma\]

• Energy: Joule \([J]=[\frac{kg\,m^2}{s^2}]\)

\[W = F \cdot d\]

• Power: Watt \([W]=[\frac{J}{s}]=[\frac{kg\,m^2}{s^3}]\)

\[P = \frac{W}{t}\]

• Pressure: Pascal \([Pa]=[\frac{N}{m^2}]=[\frac{kg}{m\,s^2}]\)

\[P = \frac{F}{A}\]

• Electric charge: Coulomb \([C]=[A s]\)

\[Q = I \cdot t\]

• Electric potential: Volt \([V]=[\frac{J}{C}]=[\frac{kg\,m^2}{A\,s^3}]\)

\[V = \frac{W}{Q}\]

• Electric resistance: Ohm \([\Omega]=[\frac{V}{A}]=[\frac{kg\,m^2}{A^2\,s^3}]\)

\[R = \frac{V}{I}\]

Uncertainty in measurement

Difference between Error and Uncertainty

• Error is the difference between a measured value and the true value. It can arise from various factors, such as limitations of measuring instruments, the nature of the physical quantity being measured, or random errors in the measurement process. Errors can be reduced by calibrating instruments, taking multiple measurements, and applying correction factors.

• Uncertainty is a measure of the range of values within which the true value of a measurement is likely to lie. It is a broader concept than error, encompassing both systematic errors (consistent, e.g., due to a faulty instrument) and random errors (variable, due to chance). Uncertainty can be estimated using statistical methods, such as the standard deviation of a set of measurements, or by using the total derivative to assess the uncertainty in a measurement due to uncertainties in the variables affecting it.

Relationship between error and uncertainty

Error refers to the specific difference between a measured value and the true value, which can be systematic (consistent, e.g., caused by an inaccurate instrument) or random (variable, due to unpredictable factors).

Uncertainty, on the other hand, is a more comprehensive concept that quantifies the range within which the true value is likely to lie, taking into account all possible sources of error (systematic and random) as well as other factors, such as variability in measurement conditions or limitations of the measurement method.

Errors contribute to uncertainty, but uncertainty is broader because it includes not only errors but also the overall imprecision of the measurement process. While errors can be minimized (e.g., through calibration or improved techniques), uncertainty is always present to some extent, as it is impossible to eliminate all sources of inaccuracy. In summary, error is a specific deviation in a measurement, while uncertainty is an estimate of the range of possible values around the measured result, reflecting the combined impact of errors and other influencing factors.

Scheme of presenting uncertainty

Quantity = Value \(\pm\) Uncertainity [Units]

Example:

Resistance = \(R \pm \Delta R = 10.0 \pm 0.1 [Ohm]\)

Uncertainty of devices and measurements

Measurement devices have a resolution, which is the smallest change in the quantity being measured that the device can detect. The uncertainty in a measurement made with a device is typically (and on our course) half of the resolution of the device.

Example 1: If a ruler has a resolution of 1 mm, by having markings every 1 mm, the uncertainty in a measurement made with the ruler is 0.5 mm. This is because the true value of the quantity being measured could be anywhere within the range of the resolution of the device.

Example 2: If the watch with markings every 1 minute shows 12:00, it means that the true time is between 11:59:30 and 12:00:30.

Example 3: Other example lets take voltmeter with resolution of 0.1 V. If the voltmeter shows 96.6 V, the true value of the voltage is between 96.55 and 96.65 V. One can write it as 96.6 V \(\pm\) 0.05 V.

Total derivative

The total derivative is a mathematical concept that describes how a function changes with respect to changes in its variables.

In the context of measurement, the total derivative can be used to quantify the so-called propagation of uncertainty in a physical quantity due to uncertainties in the variables that affect it.

\[ d f(x_1,x_2,...,x_n) = \frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2 + ... + \frac{\partial f}{\partial x_n} dx_n \]

For example for fucntion dependent on two variables, \(f(x,y)\):

\[d f(x,y) = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy \]

Uncertainty from the Total Derivative

In the context of measurement, the total derivative can be used to quantify the so-called propagation of uncertainty in a physical quantity due to uncertainties in the variables that affect it.

To estimate the uncertainty in measurement using the total derivative , you can follow these steps:

Step 1. Identify the physical quantity that you want to measure and the variables that affect it.

Step 2. Determine the uncertainties in the variables that affect the physical quantity.

Step 3. Calculate the total derivative of the physical quantity with respect to the variables that affect it.

Step 4. Use the total derivative to estimate the uncertainty in the physical quantity due to uncertainties in the variables.

For example, if a physical quantity \(f(x,y)\) is a function of variables x and y, then the total derivative of \(f\) with respect to \(x\) and \(y\) can be used to estimate the uncertainty in \(f\) due to uncertainties in \(x\) and \(y\) (i.e. \(\Delta x\) and \(\Delta y\)).

University formula

\[\Delta f(x,y) = \left| \frac{\partial f}{\partial x} \Delta x \right| + \left|\frac{\partial f}{\partial y} \Delta y \right| \]

Engineering formula (use this one!)

\[\Delta f(x,y) = \sqrt{(\frac{\partial f}{\partial x} \Delta x)^2 + (\frac{\partial f}{\partial y} \Delta y)^2} \]

Uncertainty is an unavoidable aspect of measurement in physics and other sciences. It arises from a variety of sources, including the limitations of measuring instruments, the nature of the physical quantity being measured, and the presence of random errors in the measurement process.

Example 1

Estimating Uncertainty in Volume Measurement of a Ball

The diameter of a certain ball was measured to be \(L = 42 \pm 1\) cm. Using this measurement, lets estimate the volume of the ball \(V\), and then calculate the uncertainty in this volume estimation \(\Delta V\).

Calculations of Ball’s Volume

• Diameter of the ball \[\text{Length}= L\pm \Delta L= 42 \pm 1 \text{cm}\]

• Formula for the volume of a sphere: \[V = \frac{4}{3} \pi r^3 \] where \(r\) is the radius of the ball. Since the radius \(r\) is half of the diameter, \[r = \frac{L}{2} = \frac{42}{2} = 21 \text{ cm} \] volume of the ball \(V\) can be calculated as: \[V(L) = \frac{4}{3} \pi \left(\frac{L}{2}\right)^3 = \frac{\pi L^3}{6} \approx 38808 \text{ cm}^3 \]

Estimating Volume Uncertainty

The uncertainty in the volume, \(\Delta V\), can be estimated using the propagation of uncertainty formula for functions of a single variable since the volume \(V\) depends on the radius \(r\), which in turn depends on \(L\). Note the use of absolute value here. The formula is: \[ \Delta V = \left| \frac{\partial V}{\partial L} \right| \Delta L \] where first term is partial derivative \(\frac{\partial V}{\partial L}\) and \(\Delta L = 1\) cm is the uncertainty in the diameter.

Step 1: Derive the partial derivative \(\frac{\partial V}{\partial L}\): \[ \frac{\partial V}{\partial L} = \frac{\pi}{6} (3 L^2) = \frac{\pi}{2} L^2\approx 2772 \, \text{cm}^2 \]

Step 2: Calculate \(\Delta V\): \[ \Delta V = 2772 \times 1 \approx 2772 \, \text{cm}^3 \]

Conclusion

The volume of the ball is \(38808 \, \text{cm}^3\) with an uncertainty of \(\pm 2772 \, \text{cm}^3\).

We can write is as

\[V=38808 \, \text{cm}^3 \pm 2772 \, \text{cm}^3.\]

Uncertainty as a percentage of the value

Often people write the uncertainty as a percentage of the value.

In this case it would be \[\Delta V/V \times 100 \% =2772/38808 \times 100 \% \approx 7 \% .\]

Similarly, relative uncertainty can be written as

\[\Delta V/V = 2772/38808 \approx 0.07.\]

Example 2

Calculating Resistance and Its Uncertainty

We calculate the resistance \(R\) of a resistor using Ohm’s Law, which relates resistance \(R\), voltage \(V\), and current \(I\) in a circuit: \[ R = \frac{V}{I} \]

If the voltage \(V\) and current \(I\) are measured with uncertainties \(\Delta V\) and \(\Delta I\), the uncertainty in the resistance \(\Delta R\) can be estimated using the total derivative method. \[ \text{Voltage} = V \pm \Delta V \] \[ \text{Current} = I \pm \Delta I \]

The resistance \(R\) is given by: \[ R = \frac{V}{I} \]

The uncertainty \(\Delta R\) can be calculated in two ways:

1. Simplified Linear Approximation: \[ \Delta R = \left| \frac{\partial R}{\partial V} \Delta V \right| + \left| \frac{\partial R}{\partial I} \Delta I \right| \]

2. Quadrature Method (Preferred in Engineering): \[ \Delta R = \sqrt{\left( \frac{\partial R}{\partial V} \Delta V \right)^2 + \left( \frac{\partial R}{\partial I} \Delta I \right)^2} \]

Using the formula \(R = \frac{V}{I}\), the partial derivatives are:

• With respect to \(V\): \[ \frac{\partial R}{\partial V} = \frac{\partial }{\partial V} \left( \frac{V}{I} \right)= \frac{1}{I} \]

• With respect to \(I\): \[ \frac{\partial R}{\partial I} = \frac{\partial }{\partial I} \left( \frac{V}{I} \right)= -\frac{V}{I^2} \]

Substituting the partial derivatives into the uncertainty formula, we get: \[ \Delta R = \sqrt{\left( \frac{\Delta V}{I} \right)^2 + \left( -\frac{V \Delta I}{I^2} \right)^2} \]

The resistance with its uncertainty is expressed as: \[ \text{Resistance} = R \pm \Delta R \] where: \[ R = \frac{V}{I}, \quad \Delta R = \sqrt{\left( \frac{\Delta V}{I} \right)^2 + \left( \frac{V \Delta I}{I^2} \right)^2} \]

Average and standard deviation

The average of N measurements of a physical quantity x_n is given by:

\[\bar x = \frac{1}{N} \sum_{n=1}^{N} x_n \] where \(n=1,2,3,...,N\).

The standard deviation of the N measurements is given by:

\[\Delta x = \sqrt{\frac{1}{N-1} \sum_{n=1}^{N} (x_n - \bar x)^2} \]

Mean of the measurements is the average value of the measurements, while the standard deviation is a measure of the spread or dispersion of the measurements around the mean value. The standard deviation is a measure of the uncertainty in the measurements and can be used to estimate the error in the average value.

The average and standard deviation are important statistical quantities that are used to describe the distribution of measurements and to make comparisons between different sets of data. By calculating the average and standard deviation of a set of measurements, scientists can determine the accuracy and reliability of their experimental data and make more meaningful conclusions about the physical quantity being measured.

Example of mean and standard deviation

Pizza delivery time

Consider measuring the time it takes for a pizza to be delivered to your house. You order a pizza from a local restaurant and record the delivery times for 10 different orders. The delivery times are recorded in the following table:

Measurement n	Delivery \(t_n\) [min]
1	30
2	25
3	31
4	40
5	23
6	45
7	30
8	25
9	35
10	47

Lets find the shortest and longest delivery time, the average delivery time, and the standard deviation of the delivery times.

Shortest and Longest Delivery Time:

• Shortest delivery time: 23 minutes
• Longest delivery time: 47 minutes

Average Delivery Time:

\[\small \begin{align} \text{Average delivery time} &= \frac{1}{N} \sum_{n=1}^{N} t_n\\ &= \frac{1}{10} (30 + 25 + 31 + 40 + 23 + 45 + 30 + 25 + 35 + 47)\\ &= 32.6 \text{ minutes} \end{align} \]

Standard Deviation: \[\small \begin{align} \text{Standard deviation} &= \sqrt{\frac{1}{N-1} \sum_{n=1}^{N} (t_n - \bar{t})^2}\\ &= \sqrt{\frac{(30 - 32.6)^2 + (25 - 32.6)^2 + \ldots + (47 - 32.6)^2}{9}}\\ &\approx 8.9 \text{ minutes} \end{align} \]

In conclusion, the average delivery time is 32.6 minutes, with a standard deviation of 8.9 minutes.

Error bar

Error bars are a graphical representation of the uncertainty in a measurement. They are typically shown as vertical or horizontal lines on a graph that indicate the range of values within which the true value of the measurement is likely to lie. Error bars are used to visualize the uncertainty in a measurement and to make comparisons between different data points.

Example of error bar in python

import matplotlib.pyplot as plt

# Data
x = [1, 2, 3, 4, 5]
y = [10, 15, 20, 25, 30]
y_err = [1, 2, 1.5, 2.5, 1]

# Plot with error bars
plt.errorbar(x, y, yerr=y_err, fmt='o', capsize=5)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Error Bar Example')
plt.show()

Measurement in clasical and quantum physics

Classical Physics

In classical physics, measurements are made using classical mechanics, which describes the motion of macroscopic objects such as planets, cars, and baseballs. Classical physics is based on the laws of Newtonian mechanics, which describe the motion of objects in terms of forces, masses, and accelerations.

Quantum Physics

In Quantum Mechanics, measurements are made using quantum mechanics, which describes the behavior of microscopic objects such as atoms, electrons, and photons. Quantum mechanics is based on the principles of wave-particle duality, uncertainty, and superposition.

In quantum physics, measurements are typically made using quantum instruments such as particle detectors, spectrometers, and quantum computers. These instruments are designed to measure physical quantities such as energy, momentum, position, and spin.

In Quantum Mechanics measurements are described by the wave function, which represents the probability distribution of possible measurement outcomes. The wave function is a mathematical function that describes the state of a quantum system and predicts the probabilities of different measurement results.

The Born rule is a fundamental principle of quantum mechanics that relates the wave function to the probability of measurement outcomes. According to the Born rule, the probability of measuring a particular value of a physical quantity is given by the square of the absolute value of the wave function \[\rho=|\psi|^2=\psi^*\psi\]

The uncertainty principle is another important concept in quantum mechanics that states that certain pairs of physical quantities, such as position and momentum, cannot be measured simultaneously with arbitrary precision. The uncertainty principle places a fundamental limit on the accuracy of measurements in quantum mechanics. For the position and momentum, the uncertainty principle is given by the Heisenberg uncertainty principle: \[\Delta x \Delta p \geq \frac{\hbar}{2}\]

Few nice examples of measurements in physics

Earth’s Circumference

Eratosthenes was a Greek mathematician, geographer, poet, astronomer, and music theorist who lived in the third century BC. He is famous for accurately calculating the Earth’s circumference using simple geometry.

Observation in Syene (modern-day Aswan, Egypt):

At noon on the summer solstice, the Sun was directly overhead, casting no shadow and reflecting in a well. This indicated that Syene lay on the Tropic of Cancer.

Measurement in Alexandria:

In Alexandria, north of Syene, Eratosthenes measured the Sun’s angle at noon during the summer solstice: 7°.

Distance Between Cities:

He hired a man to measure the distance between Syene and Alexandria: 800 km.

Calculation:

He assumed the Earth is a sphere and used the fraction of the Earth’s circumference that 7° represents: \[ \frac{7^\circ}{360^\circ} = \frac{800 \, \text{km}}{\text{Earth's circumference}} \]

Earth’s circumference: \[ \text{Circumference} = \frac{800 \, \text{km}}{\frac{7}{360}} = 40,000 \, \text{km} \] Earth’s radius: \[ R = \frac{40,000 \, \text{km}}{2 \pi} \approx 6370 \, \text{km} \]

Eratosthenes’ measurement was remarkably close to the actual value of the Earth’s circumference: 40,075 km.

Measuring Earth’s Mass with a Pendulum

The Earth’s mass can be calculated using a pendulum \(g = \frac{4 \pi^2 R}{T^2}\) and Newton’s law of gravitation \(mg = \frac{G m M}{R^2}\).

\[ mg = \frac{G m M}{R^2} \] Rearranging for \(M\) (Earth’s mass): \[ M = \frac{g R^2}{G} = \frac{4 \pi^2 R^2}{G} \frac{L}{T^2} \]

Given Values:

• Gravitational constant: \(G = 6.674 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\)
• Earth’s radius: \(R = 6370 \, \text{km}\)

By measuring the pendulum’s length (\(L\)) and period (\(T\)), we can calculate the Earth’s mass: \[ \text{Mass of Earth} \approx 5.97 \times 10^{24} \, \text{kg} \]

Henry Cavendish first measured Earth’s mass using his experiment to determine \(G\) in the Newtonian gravity formula.

Speed of Light Using a Microwave Oven

The speed of light, \(c\), can be measured using a microwave oven and chocolate (or cheese). Check links: here and here.

1. Remove the rotating plate and place chocolate inside the oven.
2. Heat until the chocolate starts to melt.
3. Measure the distance (\(L\)) between the melting points (one wavelength of the microwave standing wave).

Formulas:

• Wavelength: \(\lambda = 2L\)
• Speed of light: \(c = f \lambda\)

Calculations:

• Frequency of microwave oven: \(f = 2.45 \, \text{GHz}\)
• Measured distance: \(L = 0.065 \, \text{m}\) (half-wavelength) \[ c = 2.45 \times 10^9 \times 0.13 = 3.185 \times 10^8 \, \text{m/s} \]

Percentage error: \[ \frac{3.185 \times 10^8 - 2.99792458 \times 10^8}{2.99792458 \times 10^8} \times 100 \% \approx 6.3 \% \]

Geometry of the Universe

By measuring the sum of angles in a triangle, we can determine the geometry of the universe.

The sum of angles in a triangle depends on the geometry of the space:

1. Flat Space: Sum of angles = \(180^\circ\)

2. Spherical Space: Sum of angles > \(180^\circ\)

3. Hyperbolic Space: Sum of angles < \(180^\circ\).