Ed,
I do recognize that we need to be able to clip off the top of an ellipsoid as the model of choice instead of settling for a complete half. I'll begin working on that refinement . I tackled the easy problem first, as I often do. I'll also work on the math aspect and see how the ratios turn out. In terms of simplifications, w hat you propose sounds reasonable. The answer should come as problem #17. I have an area problem in mind for #16. BTW, I recently stumbled onto a Google Book on the Internet entitled "Metrical Geometry" that presents algebraic solutions to measuring geometric forms. The book was written in 1881. It is a guide to elementary Mensuration. It is a formulaic goldmine and may prove to be of considerable value to us in the search for formulas to use in tree geometry. Bob ----- Original Message ----- From: "Edward Frank" <[email protected]> To: [email protected] Sent: Thursday, March 5, 2009 9:38:06 PM GMT -05:00 US/Canada Eastern Subject: [ENTS] Re: Problem#15 Bob, Could this ratio be something as simple as the ratio such as (minor radius squared/) (major radius squared) ? Ed ----- Original Message ----- From: Edward Frank To: [email protected] Sent: Thursday, March 05, 2009 9:20 PM Subject: [ENTS] Re: Problem#15 Bob, The limitation of what is presented in problem 15 is, as I see it, that the shape of the live oak crowns are not best represented by the full half ellipsoid, a simple calculation, but as with the earlier problem dealing with the sphere, but really just the top section of this ellipse with varying proportions of the full half ellipsoid. I don't know how to calculate this exactly. The difference is that in the full half ellipsoid the the edges are essential vertical and slowly curve over to make the top. In a section representing just the top portion of the ellipsoid the angle of the edge may be pointing inward initially rather than vertically. This the calculated volume for a full half ellipsoid will over estimate the volume of the crown. I am wondering if the volume of the upper portion of the ellipse could be calculated thutly: If you calculate the volume of the top portion of a sphere using the height and length of the major axis as you did in problem 11. The ellipsoid would fit inside of this sphere For an ellipsoid shape this volume calculation will overestimate the volume. However then I am thinking the ratio of the major axis to the length of the minor axis for the ellipsoid should be proportional at all of the horizontal slices through the ellipsoid, so the volumes of an ellipsoid should be proportional to the volume of the sphere. Ed ----- Original Message ----- From: [email protected] To: [email protected] Sent: Thursday, March 05, 2009 8:58 PM Subject: [ENTS] Re: Problem#15 Larry, I believe that those huge southern live oaks that you are documenting signal the need for us to expand our set of measurements - principally for the biggest of the trees that we document. Our challenge is to choose a set of measurements that can be done in the field and that convey the size of the tree. What do you think of the area of the crown projected onto the ground? I have an idea of how best to collect the field data and run the numbers through a spreadsheet to get the area covered, but don't want to pursue development of the process if the rest of you don't see merit to the idea. I'm anxious to also hear from Ed, Will, Dale, Don, Don, et al. on the idea. Bob ----- Original Message ----- From: "Larry" <[email protected]> To: "ENTSTrees" <[email protected]> Sent: Thursday, March 5, 2009 4:41:48 PM GMT -05:00 US/Canada Eastern Subject: [ENTS] Re: Problem#15 Bob, Thanks for the formula. This is some good stuff Bob! Larry --~--~---------~--~----~------------~-------~--~----~ Eastern Native Tree Society http://www.nativetreesociety.org Send email to [email protected] Visit this group at http://groups.google.com/group/entstrees?hl=en To unsubscribe send email to [email protected] -~----------~----~----~----~------~----~------~--~---
