Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Use barycentric coordinates to recognize points inside a triangle.
- Justify the conclusion "α=1/4; all weights are positive, so P lies inside the triangle" using the stated assumptions.
Before you start
Affine combinations and coordinate geometry.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Express P=(1,1) using vertices A=(0,0), B=(4,0), C=(0,2).
Why this math matters
Use barycentric coordinates to recognize points inside a triangle. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The triangle is nondegenerate.
- Inside means the ordinary Euclidean filled triangle.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Locate a point through triangle weights
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Express P=(1,1) using vertices A=(0,0), B=(4,0), C=(0,2).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
P=αA+βB+γC with α+β+γ=1
Weights summing to one describe an affine position.
Work through the mathematics
4β=1 and 2γ=1 give β=1/4, γ=1/2
Match the coordinate components independently.
Check the conclusion
α=1/4; all weights are positive, so P lies inside the triangle
Nonnegative barycentric weights describe the closed triangle, with positive weights giving its interior.
The result
α=1/4; all weights are positive, so P lies inside the triangle
Nonnegative barycentric weights describe the closed triangle, with positive weights giving its interior.
Common mistakes to catch
- Weights must sum to one.
- A negative weight generally signals a point outside the triangle.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find the centroid weights.
Show a hint
Give each vertex equal influence.
Reveal answer and explanation
(1/3,1/3,1/3)
Their sum is one and the resulting point is the vertex average.
Practice 2
What does a zero weight imply when the other two are positive?
Show a hint
One vertex contributes nothing.
Reveal answer and explanation
The point lies on the opposite edge
It becomes a convex combination of the two remaining vertices.
Take the idea with you
Interpolate a color or temperature value specified at three mesh vertices.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Recover area and orientation from a cross product
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