Problem 84 - Calculus Single Variable

Question

Suppose \(A=\left(x_{1}, y_{1}\right)\) and \(B=\left(x_{2}, y_{2}\right)\) are any two distinct points in the plane. Let $$ \begin{array}{l} \varphi_{1}(t)=(1-t) \cdot x_{1}+t \cdot x_{2} \\ \varphi_{2}(t)=(1-t) \cdot y_{1}+t \cdot y_{2} \end{array} $$ Show that \(x=\varphi_{1}(t), \quad y=\varphi_{2}(t), \quad 0 \leq t \leq 1,\) is a para- meterization of line segment \(\overline{A B}\).

Step-by-Step Solution

Answer

This parameterization traces the line segment from \( A \) to \( B \) for \( 0 \leq t \leq 1 \).

Step 1: Understand Parameterization

Parameterization via a parameter \( t \) effectively describes a line segment using a linear combination of point coordinates. In this context, the line segment from point \( A(x_1, y_1) \) to point \( B(x_2, y_2) \) is being described using the parameter \( t \).

Step 2: Interpret the Expressions for \( \varphi_1(t) \) and \( \varphi_2(t) \)

The expressions \( \varphi_1(t) = (1-t) \cdot x_1 + t \cdot x_2 \) for \( x \)-coordinates, and \( \varphi_2(t) = (1-t) \cdot y_1 + t \cdot y_2 \) for \( y \)-coordinates are weighted averages that blend \( (x_1, y_1) \) and \( (x_2, y_2) \) as \( t \) varies from 0 to 1.

Step 3: Boundary Test for \( t = 0 \)

Setting \( t = 0 \), the expressions simplify to \( \varphi_1(0) = (1-0)x_1 + 0 \cdot x_2 = x_1 \) and \( \varphi_2(0) = (1-0)y_1 + 0 \cdot y_2 = y_1 \). Thus, the point is \( (x_1, y_1) = A \).

Step 4: Boundary Test for \( t = 1 \)

Setting \( t = 1 \), the expressions become \( \varphi_1(1) = (1-1)x_1 + 1 \cdot x_2 = x_2 \) and \( \varphi_2(1) = (1-1)y_1 + 1 \cdot y_2 = y_2 \). Thus, the point is \( (x_2, y_2) = B \).

Step 5: Intermediate Values of \( t \)

For \( 0 < t < 1 \), both \( 0 < \varphi_1(t) = (1-t)x_1 + tx_2 < 1 \) and \( 0 < \varphi_2(t) = (1-t)y_1 + ty_2 < 1 \), parameterizing points strictly between \( A \) and \( B \), thereby tracing the line segment \( \overline{AB} \).

Step 6: Conclusion of Parameterization

Through the boundary conditions and intermediate tests, \( (x, y) = (\varphi_1(t), \varphi_2(t)) \), where \( 0 \leq t \leq 1 \), defines points exclusively on the line segment between \( A \) and \( B \). Thus, the parameterization is verified.

Key Concepts

Linear Combination

Parametrizing a line involves leveraging a mathematical technique known as a linear combination. In simple terms, this concept allows us to express one variable in terms of two or more other variables. When it comes to geometry, particularly to find a line segment between two points, this becomes very useful.
For instance, consider two points on a plane, say \(A = (x_1, y_1)\) and \(B = (x_2, y_2)\). A linear combination helps us calculate any in-between point on the line segment joining \(A\) and \(B\). This is done by blending the coordinates of \(A\) and \(B\) using a parameter \(t\).
This parameter \(t\) typically ranges from 0 to 1, and as it changes, it gives us every point along the line segment from \(A\) to \(B\). The idea is to form expressions like \((1-t)x_1 + tx_2\) and \((1-t)y_1 + ty_2\) for the \(x\) and \(y\) coordinates, blending the points proportionally in a linear fashion. By doing this, we can study line segments as functions of \(t\), making the concept of parametrization intuitive and easy to apply.

Weighted Averages

Weighted averages offer a fundamental way to describe points along a line segment. When deciding how far a point is from one end of the segment to the other, we use weighted averages to assign importance or 'weight' to each of the endpoints.
The formulas \((1-t)x_1 + tx_2\) and \((1-t)y_1 + ty_2\) represent this idea precisely. Here, the weights \(1-t\) and \(t\) signify how much closer a point is to \(A\) or \(B\).
- When \(t = 0\), the weight on \(A\) is complete, resulting in the point being exactly at \(A\).
- Conversely, when \(t = 1\), the weight on \(B\) is complete instead, placing us directly at \(B\).
- For any \(0 < t < 1\), points calculated this way lie precisely between \(A\) and \(B\), rightly embodying a mix of both ends proportionate to the respective weights.
This method of understanding and applying weighted averages is crucial since it makes the concept of a line segment tangible as a continuous set of linear interpolations between designated endpoints.

Boundary Conditions

Boundary conditions are essential when discussing parametrization as they validate that the formulas behave as expected at the extremes of the parameter range. In the context of the line segment \(\overline{AB}\), we explore \(t\) such that \(0 \leq t \leq 1\).
The boundary conditions check:

• At \(t = 0\), we substitute into our expressions to see \(\varphi_1(0) = x_1\) and \(\varphi_2(0) = y_1\), placing the point exactly at \(A\).
• Similarly, at \(t = 1\), we find \(\varphi_1(1) = x_2\) and \(\varphi_2(1) = y_2\), confirming that the point sits perfectly at \(B\).

This verification ensures our parameterization covers both ends of the segment correctly. In between these bounds, every possible \(t\) maps to a unique point on the segment, thus establishing a clear path between \(A\) and \(B\). It affirms the correctness of our approach and guarantees that no part of the segment is unaccounted for.

Plane Geometry

Plane geometry provides the foundational context in which we analyze line segments on a two-dimensional surface. Here, line segments are more than just connections between dots -- they are entities that can be analyzed, calculated, and applied in various geometric contexts.
Understanding the fundamentals of plane geometry is critical for handling tasks like parametrizing line segments. It leverages the coordinate system, providing a grid in which points \((x, y)\) can be expressed precisely and relationships between points can be defined using geometrical functions.
Using parametrization, we craft a path using a parameter \(t\), allowing movement along this line from point \(A(x_1, y_1)\) to point \(B(x_2, y_2)\). This simplifies complex geometric analyses into manageable segments, each describable through linear functions.
Insights gleaned from plane geometry, when combined with computations like parametrization, enable us to solve a wide range of problems involving shapes, distances, and other relevant geometric entities, ultimately empowering us to explore and understand spatial relationships vividly and precisely.