Linear interpolation Definitions and Examples
In mathematics, interpolation is a method of constructing new data points within the range of a discrete set of known data points. It is often required when an exact data point is not available for the desired output, such as in certain graphing or tabulation applications. There are many types of interpolation method, but all fall into one of two categories: methods that are based on a known mathematical function or relationships between variables, and those that construct a new data point using some type of averaging technique. In this blog post, we will explore linear interpolation, including its definition, formulas, and examples.
Linear Interpolation Formula
Linear interpolation is a mathematical process for determining unknown values that lie between known values. The linear interpolation formula is used to calculate the unknown y-coordinates of a point on a line when only the x-coordinate is known, or vice versa.
To use the linear interpolation formula, one must first know the coordinates of at least two points on the line in question. These can be any two points, as long as their x-coordinates are different. Once these coordinates are known, the linear interpolation formula can be used to calculate the y-coordinate of any other point on the line.
The linear interpolation formula is:
y = y1 + (x – x1)(y2 – y1)/(x2 – x1)
where:
y = the unknown y-coordinate
y1 and y2 = the known y-coordinates of two points on the line
x = the unknown x-coordinate (or vice versa)
x1 and x2 = the known x-coordinates of two points on the line
What is Linear Interpolation Formula?
Linear interpolation is a mathematical method that can be used to estimate values between two known points. The linear interpolation formula is:
y = y0 + (x-x0)*(y1-y0)/(x1-x0)
where,
y = the estimated value
y0 = the first known value
y1 = the second known value
x0 = the first known x-value
x1 = the second known x-value
Linear interpolation
Linear interpolation is a mathematical process used to estimate values between two known points. In essence, it “fills in the gaps” between those points. The method is used extensively in scientific and engineering calculations because it is relatively simple and accurate.
There are many different ways to interpolate values, but linear interpolation is perhaps the most common. It works by drawing a straight line between two known points and then calculating the value of the desired point based on that line.
For example, let’s say you have two points on a graph: (1,2) and (3,6). You can use linear interpolation to estimate the value of the point (2,4), which lies between those two points. To do so, you would draw a straight line between (1,2) and (3,6) and then find the point on that line that corresponds to x=2. This point would have a y-value of 4, which is your estimation for the value of the desired point.
As another example, let’s say you want to know what temperature it will be at noon today but all you know is that it was 20 degrees at 6am and 30 degrees at 12pm. Using linear interpolation, you can estimate that it will be 25 degrees at noon.
Linear interpolation is relatively easy to understand and use but it does have some limitations.
Examples Using Linear Interpolation Formula
Linear interpolation is a mathematical process used to estimate values between two known points. This technique is often used in computer graphics and animation to create smooth transitions between keyframes.
Here are a few examples of how linear interpolation can be used:
1. Estimating the value of a function at a specific point:
Suppose we have a function f(x) that is continuous and differentiable on the interval [a,b]. We want to estimate the value of f(x) at some point x0 that is between a and b. We can do this by using the linear interpolation formula:
f(x0) = f(a) + (x0-a)*[f(b)-f(a)]/(b-a)
2. Creating an animated character:
One common use of linear interpolation in computer graphics is for creating animated characters. For example, suppose we want to animate a character walking across a screen from left to right. We can do this by creating keyframes at different points along the character’s path, and then using linear interpolation to fill in the frames in between.
3. Smoothing data points:
Another common use of linear interpolation is for smoothing data points. For example, suppose we have a set of data points that represent the monthly sales figures for a company over the course of a year.
Example 2: Calculate the estimated height of the boy in the fourth position.
Assuming that the boy in the fourth position is of average height for his age, we can estimate his height by linearly interpolating between the heights of the boys in the third and fifth positions. Based on the information given, we know that the boy in the third position is 1.2 meters tall, and that the boy in the fifth position is 1.6 meters tall. Therefore, we can estimate that the boy in the fourth position is approximately 1.4 meters tall.
Example 3: Find the value of y if x = 8 and some set of values are given as (5, 3.5), (10, 6)?
Assuming that the set of values given is a function of x, then the value of y when x = 8 can be found using linear interpolation.
When using linear interpolation, one first finds the two points on the graph that surround the unknown x-value. In this case, those points are (5, 3.5) and (10, 6). Then, one draws a line between those two points. The value of y can then be found by finding where that line intersects with the y-axis.
In this particular example, the line would intersect the y-axis at approximately 4.7. Therefore, the value of y when x = 8 is approximately 4.7.
What is linear interpolation?
Linear interpolation is a mathematical method for finding a value between two given points on a line. It is often used to find an estimate for a value that lies between two known values, or to find the equation of a straight line when only two points on the line are known.
Linear interpolation is based on the idea of finding a straight line that best fits two given points. The line is then extended to find the point that lies halfway between the two given points. This process can be continued to find any number of intermediate points.
To understand linear interpolation, it is first necessary to understand what a function is. A function is a set of ordered pairs (x, y) where each x corresponds to a unique y. In other words, a function takes an input (x) and produces an output (y). A graph of a function will show how the output y changes as the input x changes.
The simplest kind of function is a linear function. A linear function has the form y =mx + b, where m is the slope of the line and b is the y-intercept. The slope of a line tells us how steep the line is, and the y-intercept tells us where the line crosses the y-axis.
How to use linear interpolation
Linear interpolation is a mathematical method used to estimate values between two known points. The simplest form of linear interpolation is called “linear extrapolation”, which consists of extending a line from two known points. However, this method can lead to inaccurate results if the data is not linear.
A more sophisticated form of linear interpolation is “curvilinear interpolation”. This method uses a curve that best fits the data points, instead of a straight line. This results in greater accuracy, but is more complex to calculate.
Linear interpolation can be used for many applications, such as estimating prices between two known points, or predicting future values based on past data.
Examples of linear interpolation
Linear interpolation is a mathematical process used to estimate values between two known points. Linear interpolation is often used with time-based data, such as stock prices, where it can be used to predict future values.
There are many different ways to interpolate data, but linear interpolation is the most common and simplest method. Linear interpolation works by drawing a straight line between two known points and then estimating the value of the desired point based on that line.
To better understand linear interpolation, let’s look at some examples.
Example 1:Estimating Stock Prices
Let’s say you have a stock that you purchased for $100 per share and it is now worth $120 per share. You also know that another similar stock is currently trading at $110 per share. Using linear interpolation, you can estimate that your stock will be worth $115 per share in the future.
Example 2: Estimating Population Growth
You can also use linear interpolation to estimate population growth. For example, let’s say you know that a city had a population of 10,000 people in 2010 and 15,000 people in 2015. Using linear interpolation, you can estimate that the city’s population will be 12,500 people in 2020.
Pros and Cons of Linear Interpolation
Linear interpolation is a mathematical method used to estimate values between two known points. It is commonly used in finance and economics when trying to predict future values based on past data.
There are both pros and cons to using linear interpolation. Some of the advantages include that it is simple to understand and implement, as well as being relatively accurate. On the other hand, one of the main disadvantages is that it can be biased if the data points are not evenly spaced.
Conclusion
Linear interpolation is a mathematical method that allows us to estimate values between two known points. This technique is commonly used in data analysis and graphing, and can be a helpful tool for students and professionals in a variety of fields. Interpolation can be used to predict future trends, fill in missing data points, or simply to better understand the relationships between variables. No matter how it’s being used, linear interpolation is a powerful tool that can provide insights that would otherwise be difficult or impossible to obtain.
