In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C and D on a line, their cross ratio is defined as where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.)The point D is the harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is −1, called the harmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name anharmonic ratio. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry. The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century. Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio. (Wikipedia).
👉 Learn how to solve proportions. Two ratios are said to be proportional when the two ratios are equal. Thus, proportion problems are problems involving the equality of two ratios. When given a proportion problem with an unknown, we usually cross-multiply the two ratios and then solve for
From playlist Solve Proportions
Solving a proportion using the cross product ex 7, 8/5 = (4/3x)
👉 Learn how to solve proportions. Two ratios are said to be proportional when the two ratios are equal. Thus, proportion problems are problems involving the equality of two ratios. When given a proportion problem with an unknown, we usually cross-multiply the two ratios and then solve for
From playlist How to Solve a Proportion
Proportions - Intro, Cross Product, and Applications
This video defines a proportion and then solves several problems using cross products.
From playlist Number Sense - Decimals, Percents, and Ratios
Using cross product to verify a proportion
👉 Learn how to solve proportions. Two ratios are said to be proportional when the two ratios are equal. Thus, proportion problems are problems involving the equality of two ratios. When given a proportion problem with an unknown, we usually cross-multiply the two ratios and then solve for
From playlist Solve Proportions
This video introduces the Golden ratio and provides several examples of where the Golden ratio appears. http:mathispower4u.com
From playlist Mathematics General Interest
Ratios Introduction - what are ratios?
Ratios are used to compare different quantities. In this introduction to ratios we will look at what ratios are, how we deal with ratios of different measurement units and that ratios can be simplified. To donate to the tecmath channel:https://paypal.me/tecmath To support tecmath on Pa
From playlist Ratios
What is an angle and it's parts
👉 Learn how to define angle relationships. Knowledge of the relationships between angles can help in determining the value of a given angle. The various angle relationships include: vertical angles, adjacent angles, complementary angles, supplementary angles, linear pairs, etc. Vertical a
From playlist Angle Relationships
Identifying and solving proportions
This is a short video tutorial on identifying and solving proportions. For interactive applets, worksheets, and more videos go to http://www.mathvillage.info
From playlist All about ratios and proportions
This video defines a proportion and explains how to solve a proportion. http://mathispower4u.wordpress.com/
From playlist Proportions
Genetics - Mendelian Experiments - Monohybrid and Dihybrid Crosses - Lesson 3 | Don't Memorise
Crosses in genetics can be presented theoretically in more than one ways. One of the most simple methods of presenting a Cross is using the Punnett Square method. The technique has several advantages, especially when it comes to complex calculations of Dihybrid crosses. Want to know how th
From playlist Genetics - Basics
Introduction to Projective Geometry via Tic-Tac-Toe Grids #SoME2
My entry for @3blue1brown's Summer of Math Exposition 2022. It's my first video ever and there are a million things I would like to improve. Mostly the audio quality and my terrible, terrible accent. 😅 Alas, perfect is the enemy of the good, isn't it. So, here we go. #SoME2 ------------
From playlist Summer of Math Exposition 2 videos
Geometry of Growth and Form: Commentary on D'Arcy Thompson | John Milnor
John Milnor, Co-Director of the Institute for Mathematical Sciences at Stony Brook University http://www.math.sunysb.edu/~jack September 24, 2010 In this lecture, John Milnor, Co-Director of the Institute for Mathematical Sciences at Stony Brook University and a former member of the Facul
From playlist Mathematics
Monohybrid Inheritance and Codominance | Revision for Biology A Level and IB
I want to help you achieve the grades you (and I) know you are capable of; these grades are the stepping stone to your future. Even if you don't want to study science or maths further, the grades you get now will open doors in the future Tutoring - We can match you with an experienced tut
From playlist AQA A-Level Biology | Ultimate Revision Playlist
Gene Linkage | Revision for Biology A-Level and IB
I want to help you achieve the grades you (and I) know you are capable of; these grades are the stepping stone to your future. Even if you don't want to study science or maths further, the grades you get now will open doors in the future. Tutoring - We can match you with an experienced t
From playlist AQA A-Level Biology | Ultimate Revision Playlist
Genetics - Mendelian Experiments - Lesson 2 | Don't Memorise
Gregor Mendel's work was a breakthrough in the field of Biology, but how did Mendel carry out his experiments? How did he obtain results? Just crossing plants with so many characters was not an easy job. To simplify things, Mendel came up with the concept of Monohybrid cross. Want to know
From playlist Genetics - Basics
One of the most famous free kicks ever taken - but how far from the goal was Brazil's Roberto Carlos? More links & stuff in full description below ↓↓↓ Discussed by Federico Ardila from San Francisco State University. More Federico videos: http://bit.ly/Federico_Numberphile More football/
From playlist Football (soccer) on Numberphile
Genetics - Understanding Crosses - Lesson 6 | Don't Memorise
Studying Laws of Genetics is not as easy as it seems. How do we determine the Genotype of a given organism? What is the Genotypic Ratio obtained after a Dihybrid cross? How exactly are Gametes formed and how many combinations of Gametes are obtained during a Trihybrid Cross? Do you think y
From playlist Genetics - Basics
Let’s Solve This Proportion…Step-by-Step…
TabletClass Math: https://tcmathacademy.com/ Math help with solving a proportion. Also, math help with ratios, rates and cross product. For more math help to include math lessons, practice problems and math tutorials check out my full math help program at https://tcmathacademy.com/ M
From playlist GED Prep Videos
This video defines a ratio and provides several examples on how to write a ratio and shows how to simplify a ratio. http://mathispower4u.wordpress.com/
From playlist Ratios and Rates